††thanks: $\dagger$ Corresponding author: A. Acus

# Logarithm of multivector in real 3D Clifford algebras

A. Acus  Institute of Theoretical Physics and Astronomy,Vilnius
University,Saulėtekio 3, LT-10257 Vilnius, Lithuania<EMAIL_ADDRESS>A. Dargys  Center for Physical Sciences and Technology, Saulėtekio 3, LT-10257
Vilnius, Lithuania<EMAIL_ADDRESS>
###### Abstract.

Closed form expressions for a logarithm of general multivector (MV) in base-
free form in real geometric algebras (GAs) $\mathit{Cl}_{p,q}$ are presented
for all $n=p+q=3$. In contrast to logarithm of complex numbers (isomorphic to
$\mathit{Cl}_{0,1}$), 3D logarithmic functions, due to appearance of two
double angle arc tangent functions, allow to include two sets of sheets
characterized by discrete coefficients. Formulas for generic and special cases
of individual blades and their combinations are provided.

###### Key words and phrases:

Clifford (geometric) algebra, logarithms of Clifford numbers, computer-aided
theory

###### 1991 Mathematics Subject Classification:

Primary 15A18; Secondary 15A66

##  1\. Introduction

Logarithm properties are well-known for real and complex numbers. Except the
Hamilton quaternions which are isomorphic to $\mathit{Cl}_{0,2}$, the
properties of logarithm in other 2D algebras (some partial formulas for 2D GAs
are provided in [1, 2, 3, 4]) and higher dimensional Clifford algebras remain
uninvestigated as yet. In general, GA logarithm properties are simplest for
anti-Euclidean algebras $\mathit{Cl}_{0,n}$ [5, 6]. As in the complex algebra
case we expect at least to have a principal logarithm and a part that makes
the GA logarithm a multivalued function.

Recently in papers [7, 8], which will be the starting point for the present
article, we have performed a detailed investigation of 3D exponential
functions in real GAs. However, the GA logarithm is more difficult to analyze
since one must take into account the multi-valuedness and the fact that in 3D
algebras (except $\mathit{Cl}_{0,3}$) the logarithm may not exist for all MVs.
Here, we have treated the logarithm as an inverse problem using for this
purpose the Mathematica symbolic package, more precisely as an inverse GA
function to exponential in separate 3D algebras $\mathit{Cl}_{0,3}$,
$\mathit{Cl}_{3,0}$, $\mathit{Cl}_{1,2}$, and $\mathit{Cl}_{2,1}$. The final
GA logarithm formulas were checked symbolically as well as numerically. They
are in complete agreement with more general formulas [9] suitable for
computation on any function of diagonalizable multivector (MV). The exact
logarithm formulas also have been applied to study convergence of series
expansion MV logarithms.

In Sec. 2 the notation is introduced. Since the logarithm is closely related
with a two argument arc tangent function $\arctan(x,y)$, its properties are
summarized in this section as well. In Sec. 3 the logarithm of the simplest,
namely $\mathit{Cl}_{0,3}$ algebra is considered. Since algebras
$\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$ are isomorphic, in Sec. 4 the
logarithms for both algebras are investigated simultaneously. In Sec. 5 the
logarithm of the most difficult $\mathit{Cl}_{2,1}$ algebra is presented.
Since GA logarithm may be applied to GA square root calculation [4, 10], in
Sec. 6 the roots and arbitrary fractional powers of MV are discussed. In Sec.
7 the relations of the logarithm to GA inverse trigonometric and hyperbolic
functions are presented. Finally, in Sec. 8 we summarize the obtained results.

##  2\. Notation and general properties of GA logarithm

A general MV in GA space is expanded in the orthonormal basis in inverse
degree lexicographic ordering:
$\\{1,\mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3},\mathbf{e}_{12},\mathbf{e}_{13},\mathbf{e}_{23},\mathbf{e}_{123}\equiv
I\\}$, where $\mathbf{e}_{i}$ are basis vectors, $\mathbf{e}_{ij}$ are the
bivectors and $I$ is the pseudoscalar. The number of subscripts indicates the
grade. The scalar is a grade-0 element, the vectors $\mathbf{e}_{i}$ are the
grade-1 elements, etc. In the orthonormalized basis the geometric products of
basis vectors satisfy the anticommutation relation,
$\mathbf{e}_{i}\mathbf{e}_{j}+\mathbf{e}_{j}\mathbf{e}_{i}=\pm 2\delta_{ij}$.
For $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{0,3}$ algebras the squares of basis
vectors, correspondingly, are $\mathbf{e}_{i}^{2}=+1$ and
$\mathbf{e}_{i}^{2}=-1$, where $i=1,2,3$. For mixed signature algebras such as
$\mathit{Cl}_{2,1}$ and $\mathit{Cl}_{1,2}$ the squares are
$\mathbf{e}_{1}^{2}=\mathbf{e}_{2}^{2}=1$, $\mathbf{e}_{3}^{2}=-1$ and
$\mathbf{e}_{1}^{2}=1$, $\mathbf{e}_{2}^{2}=\mathbf{e}_{3}^{2}=-1$,
respectively.

The general MV in real Clifford algebras $\mathit{Cl}_{p,q}$ for $n=p+q=3$ is

$\begin{split}\mathsf{A}=&\,a_{0}+a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}+a_{12}\mathbf{e}_{12}+a_{23}\mathbf{e}_{23}+a_{13}\mathbf{e}_{13}+a_{123}I\\\
\equiv&\,a_{0}+\mathbf{a}+\mathcal{A}+a_{123}I=a_{0}+\mathsf{A}_{1,2}+a_{123}I=\mathsf{A}_{0,1,2,3},\end{split}$
(2.1)

where $a_{i}$, $a_{ij}$ and $a_{123}$ are the real coefficients, and
$\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}$ and
$\mathcal{A}=a_{12}\mathbf{e}_{12}+a_{23}\mathbf{e}_{23}+a_{13}\mathbf{e}_{13}$
is, respectively, the vector and bivector. $I$ is the pseudoscalar,
$I=\mathbf{e}_{123}$. The double index in MV $\mathsf{A_{i,j}}$ indicates the
sum of MVs of grades $i$ and $j$, i.e.
$\mathsf{A_{i,j}}=\langle\mathsf{A}\rangle_{i}+\langle\mathsf{A}\rangle_{j}$.

The main involutions, namely the reversion, grade inversion and Clifford
conjugation denoted, respectively, by tilde, circumflex and their combination
are defined by

$\begin{split}&\widetilde{\mathsf{A}}=a_{0}+\mathbf{a}-\mathcal{A}-a_{123}I,\quad\wideparen{\mathsf{A}}=a_{0}-\mathbf{a}+\mathcal{A}-a_{123}I,\\\
&\widetilde{\wideparen{\mathsf{A}}}=a_{0}-\mathbf{a}-\mathcal{A}+a_{123}I.\end{split}$
(2.2)

###  2.1.  General properties of GA logarithm

The logarithm of MV is another MV that belongs to the same geometric algebra
(GA). The defining equation for MV logarithm is
$\log(\mathrm{e}^{\mathsf{A}})=\mathsf{A}$, where
$\mathsf{A}\in\mathit{Cl}_{p,q}$. The GA logarithm is a multivalued function.
In [1] it was suggested that ”The principal value of the logarithm can be
defined as the MV $\mathsf{M}=\log(\mathsf{Y})$ with the smallest norm”, where
$Y\in\mathit{Cl}_{p,q}$. The natural norm for a MV is the determinant norm
defined in subsection 2.2. The following properties hold for MV logarithm:

$\begin{split}&\log(\mathsf{A}\mathsf{B})=\log(\mathsf{A})+\log(\mathsf{B})\quad\text{if\
}\mathsf{A}\mathsf{B}=\mathsf{B}\mathsf{A},\\\
&\mathrm{e}^{\log(\mathsf{A})}=\mathsf{A},\quad\mathrm{e}^{-\log(\mathsf{A})}=\mathsf{A}^{-1},\\\
&\widetilde{\log(\mathsf{A})}=\log(\widetilde{\mathsf{A}}),\quad\wideparen{\log(\mathsf{A})}=\log(\wideparen{\mathsf{A}}),\quad\widetilde{\wideparen{\log(\mathsf{A})}}=\log(\widetilde{\wideparen{\mathsf{A}}}),\\\
&\mathsf{V}\,\log(\mathsf{A})\mathsf{V}^{-1}=\log(\mathsf{V}\mathsf{A}\mathsf{V}^{-1}).\end{split}$
(2.3)

In the last expression the transformation $\mathsf{V}$, for example the rotor,
is pushed inside the logarithm.

###  2.2.  GA logarithm series

In analogy with a definition of logarithm in complex plane for GA logarithm we
can write

$\log\mathsf{A}=\sum_{k=1}^{\infty}\frac{(-1)^{k-1}(\mathsf{A}-1)^{k}}{k},\quad\mathrm{if}\quad|\mathsf{A}-1|<1,$
(2.4)

where $|\mathsf{A}-1|$ denotes the determinant norm. For arbitrary MV the
determinant norm is defined as an absolute value of determinant
$\operatorname{Det}(\mathsf{B})$ of MV $\mathsf{B}$ raised to fractional power
$1/k$, where $k=2^{\lceil n/2\rceil}$, i.e.,
$|\mathsf{B}|=\bigr{(}\operatorname{Det}(\mathsf{B})\bigl{)}^{1/k}>0$. For
algebras having negative determinant instead the semi-norm (aka pseudoscalar)
is introduced
$\lvert\mathsf{B}\rvert=\bigr{(}\text{abs}(\operatorname{Det}(\mathsf{B}))\bigl{)}^{1/k}\geq
0$. The equality sign means that in case of semi-norm the determinant may be
zero although $\mathsf{B}\neq 0$. In the following the same symbol will be
used for both the norm and semi-norm. The (semi-)norm can be interpreted as a
number of multipliers needed to define $\operatorname{Det}(\mathsf{B})$. In 3D
algebras ($n=3$) we have $k=2^{\lceil 3/2\rceil}=2^{2}=4$, which is the degree
of characteristic [11] polynomial $\operatorname{Det}(\mathsf{B})$. In this
way found integer $k$ coincides with the number of multipliers in the 3D
determinant:
$\operatorname{Det}(\mathsf{B})=\mathsf{B}\widetilde{\mathsf{B}}\wideparen{\mathsf{B}}\wideparen{\widetilde{\mathsf{B}}}$.
The determinant norm for MV $\mathsf{B}$ in 3D algebras, therefore, is
$|\mathsf{B}|=\sqrt[4]{\mathop{\mathrm{abs}}(\operatorname{Det}(\mathsf{B}))}$
. It can be shown that for any GA that holds a basis element with property
$\mathbf{e}_{i}^{2}=-1$ by adding a scalar one can construct a MV the norm of
which may be identified with a module of a complex number. For example in
$\mathit{Cl}_{3,0}$ the norm of $\mathsf{B}=1+\mathbf{e}_{12}$ is
$\sqrt{(1+\mathbf{e}_{12})(1-\mathbf{e}_{12})}=\sqrt{2}$ which coincides with
$|\mathsf{B}|=\sqrt[4]{\mathop{\mathrm{abs}}(\operatorname{Det}(\mathsf{B}))}=\sqrt{2}$
. (also refer to Example 1 below).

If the MV has a numerical form, to minimize the number of multiplications it
is convenient to represent the logarithm in a nested form (aka Horner’s rule).
The logarithmic series [12] (also called Mercator series), if rewritten
according to Horner’s rule, assumes the following form,

$\begin{split}\log\mathsf{B}=\mathsf{B}(1+\mathsf{B}(-\tfrac{1}{2}+\mathsf{B}(\tfrac{1}{3}+\mathsf{B}(-\tfrac{1}{4}+\mathsf{B}(\tfrac{1}{5}+\dotsm))))),\quad\textrm{where\
}\mathsf{B}=\mathsf{A}-1.\end{split}$ (2.5)

Example 1. MV equivalent to complex number. Let’s take the MV
$\mathsf{A}=\frac{9}{10}-\frac{1}{3}\mathbf{e}_{3}$ the determinant norm of
which in $\mathit{Cl}_{0,3}$ is
$|\mathsf{B}|=|\mathsf{A}-1|=\frac{\sqrt{109}}{30}\approx 0.34801<1$.
Therefore, the standard series, Eq. (2.4), may be applied to find an
approximate value (the result found by exact formula in Example 2 is
$\log\frac{\sqrt{829}}{30}-\mathbf{e}_{3}\arctan\frac{10}{27}\approx-0.0410873-\mathbf{e}_{3}0.354706$).
Since $\mathbf{e}_{3}^{2}=-1$ and it is the only basis vector in the
considered MV, one may replace the MV by complex number
$z=\frac{9}{10}-\mathrm{i}\frac{1}{3}$. The module is
$|z-1|=\frac{\sqrt{109}}{30}$ which coincides with the MV determinant norm.
Then, $\log z\approx-0.0410873-\mathrm{i}\,0.354706$.

Now let’s calculate the logarithm of
$\mathsf{A}^{\prime}=-\frac{9}{10}-\frac{1}{3}\mathbf{e}_{3}$ by the Horner
series (2.5). Since $|\mathsf{A}^{\prime}-1|=\frac{\sqrt{3349}}{30}\approx
1.92902>1$ the series diverges. As shown in Example 2, the logarithm can be
easily computed if exact GA logarithm formula obtained in the present paper is
used. After replacement of the MV by complex number we obtain that
$|z^{\prime}-1|=1.92902$ which again coincides with the module of
$z^{\prime}=-\frac{9}{10}-\frac{1}{3}\mathrm{i}$. Computing the value of
logarithm by Mathematica command
$\mathrm{FunctionExpand}[\mathrm{Log}[z^{\prime}]]$ we obtain $\log
z^{\prime}=\mathrm{i}(-\pi+\arctan(10/27))-\log(30)+\log(829)/2$ which has the
same numerical value as shown in the Example 2.

###  2.3. Double-argument arc tangent function

Figure 1. Graphical representation of single
$\arctan(y/x)=\arctan(\sin\theta/\cos\theta)$ (dashed line) and double
$\arctan(x,y)=\arctan(\cos\theta,\sin\theta)$ (solid line) argument tangent
functions used by Mathematica. $\theta$ is an angle between $x$ axis and
vector (not shown) attached to the center of complex $x-y$ plane. The vector
may be rotated from $x$-axis anticlockwise, $\theta=(0...\pi]$, or clockwise,
$\theta=[0...-\pi)$ as shown by arrows in the inset. In the inset also the
numbering of the quadrants 1-4 and the branching, represented by thick line on
the negative part of $x$ axis, are shown.

GA logarithm as we shall see, in its nature is a multi-valued function with
period $2\pi$. To account for quadrant sign in complex plane properly we shall
need the double argument arc tangent function as given in the Mathematica, the
properties of which are briefly mentioned below. Figure 1 shows the single and
double argument arc tangent functions. The former has period $\pi$ and its
principal values lie in the interval $\theta=[-\pi/2,\pi/2)$, while the double
argument arc tangent has, respectively, $2\pi$ period and principal values in
$\theta=[-\pi,\pi)$. The inset on the right side of Fig. 1 shows the quadrants
1-4 in $x-y$ plane. Note that the anticlockwise rotation is done from quadrant
1 to quadrant 2, while clockwise rotation in order 3$\to$4, so that a jump in
the double arc tangent value and associated branching occurs on the negative
side of $x$-axis rather than on $y$ axis as is in the standard single argument
case. Also, in Fig. 1 note small points on vertical branching steps that
indicate that respective arc tangent value on periodic line belong to upper
rather than lower part, i.e., at $\theta=\pi$ we have
$\arctan(\cos(\pi),\sin(\pi))=\arctan(-1,0)=\pi$, however, after addition of
infinitesimal angle $\arctan(\cos(\pi+0_{+}),\sin(\pi+0_{+}))=-\pi$.
Similarly, at $\theta=-\pi$ we have $\arctan(\cos(-\pi),\sin(-\pi))=\pi$, and,
$\arctan(\cos(-\pi+0_{+}),\sin(-\pi+0_{+}))=-\pi$. If $x,y$ were replaced by
real numbers Mathematica will switch automatically to a single argument arc
tangent in the first quadrant and principal values, for example,
$\arctan(17,10)=\arctan(10/17)$, $\arctan(-17,10)=\pi-\arctan(10/17)$,
$\arctan(17,-10)=-\arctan(10/17)$, $\arctan(-17,-10)=-\pi+\arctan(10/17)$.

In the terms of a standard arc tangent function the argument of which is in
the range $(-\pi/2,\pi/2)$, the double tangent principal values now in the
range $(-\pi,\pi)$ can be expressed as follows:

$\arctan(x,y)=\begin{cases}\arctan({\frac{y}{x}})&{\text{if }}x>0,\\\
\arctan({\frac{y}{x}})-\pi&{\text{if }}x<0{\text{ and }}y\geq 0,\\\
\arctan({\frac{y}{x}})+\pi&{\text{if }}x<0{\text{ and }}y<0,\\\
+{\frac{\pi}{2}}&{\text{if }}x=0{\text{ and }}y>0,\\\
-{\frac{\pi}{2}}&{\text{if }}x=0{\text{ and }}y<0,\\\
\textrm{undefined}&{\text{if }}x=0{\text{ and }}y=0\,.\end{cases}$ (2.6)

We will start from $\mathit{Cl}_{0,3}$ where the expanded exponential in a
coordinate-form has the simplest MV coefficients and the logarithm exists for
all MVs.

##  3\.  MV logarithms in $\mathit{Cl}_{0,3}$

###  3.1.  Logarithm formula for generic MV

The term ”generic” here will be understood as ”not creating the problems”. If
for a given set of MV coefficients the generic formula is not applicable, for
example, due to nullification of a denominator, or due to appearance of an
undefined subexpression like $\arctan(0,0)$, we will refer to it as ”special
case”. Special cases will be covered by more elaborate formula later.

######  Theorem 3.1 (Logarithm of multivector in $\mathit{Cl}_{0,3}$).

The generic logarithm of MV $\mathsf{A}=a_{0}+\mathbf{a}+\mathcal{A}+a_{123}I$
is the MV given by

$\log(\mathsf{A})=\frac{1}{2}\bigl{(}\mathsf{A}_{{0}_{+}}+\mathsf{A}_{{0}_{-}}+\mathsf{A}_{{1,2}_{+}}+\mathsf{A}_{{1,2}_{-}}+(\mathsf{A}_{{0}_{+}}-\mathsf{A}_{{0}_{-}})I\bigr{)},$
(3.1)

with

$\displaystyle\mathsf{A}_{{0}_{+}}$
$\displaystyle=\frac{1}{2}\log\bigl{(}(a_{0}+a_{123})^{2}+a_{+}^{2}\bigr{)},$
$\displaystyle a_{+}\neq 0,$ (3.2) $\displaystyle\mathsf{A}_{{0}_{-}}$
$\displaystyle=\frac{1}{2}\log\bigl{(}(a_{0}-a_{123})^{2}+a_{-}^{2}\bigr{)},$
$\displaystyle a_{-}\neq 0,$ (3.3) $\displaystyle\mathsf{A}_{{1,2}_{+}}$
$\displaystyle=\frac{1}{a_{+}}\big{(}\arctan(a_{0}+a_{123},a_{+})+2\pi
c_{1+}\big{)}\bigl{(}1+I\bigr{)}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},$
$\displaystyle a_{+}\neq 0,$ (3.4) $\displaystyle\mathsf{A}_{{1,2}_{-}}$
$\displaystyle=\frac{1}{a_{-}}\big{(}\arctan(a_{0}-a_{123},a_{-})+2\pi
c_{1-}\big{)}\bigl{(}1-I\bigr{)}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},$
$\displaystyle a_{-}\neq 0.$ (3.5)

The MVs $\mathsf{A}_{{0}_{\pm}},\mathsf{A}_{{1,2}_{\pm}}$ and
$\mathsf{A}_{{0}_{\pm}}I$ denote, respectively, the scalar,
vector$\pm$bivector and the pseudoscalar components.
$c_{1_{\pm}},c_{2_{\pm}}\in\mathbb{Z}$ are arbitrary integers. The scalars
$a_{+}\geq 0$ and $a_{-}\geq 0$ are given by expressions [7, 8],

$\displaystyle a_{-}=$
$\displaystyle\sqrt{-(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})+2I\mathbf{a}\mathbin{\wedge}\mathcal{A}}$
(3.6) $\displaystyle=$
$\displaystyle\sqrt{(a_{3}+a_{12})^{2}+(a_{2}-a_{13})^{2}+(a_{1}+a_{23})^{2}},$
$\displaystyle a_{+}=$
$\displaystyle\sqrt{-(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})-2I\mathbf{a}\mathbin{\wedge}\mathcal{A}}$
(3.7) $\displaystyle=$
$\displaystyle\sqrt{(a_{3}-a_{12})^{2}+(a_{2}+a_{13})^{2}+(a_{1}-a_{23})^{2}}\,,$

Proof. It is enough to check that after substitution of (3.1) into GA
exponential formula (1) of [7] one will get the initial MV $\mathsf{A}$.

The Theorem 3.1 gives the GA logarithm in a basis-free form. However, the
derivation of above given generic logarithm formula at first was done in a
coordinate form, from which the Theorem 3.1 follows (see Appendix A). The
Theorem 3.1 ensures the existence of GA logarithm for all MVs with real
coefficients in $\mathit{Cl}_{0,3}$, because in the mentioned algebra the zero
determinant of MV ($\operatorname{Det}\mathsf{A}=0$) occurs only if
$\mathsf{A}=0$. As we shall see this property does not hold for remaining
algebras.

###  3.2.  Special cases

In Theorem 3.1 it was presumed that the both scalars $a_{-}$ and $a_{+}$ do
not vanish. This assumption is equivalent to the condition that the sum of
vector and bivector must have non-zero-determinant111We shall remind that for
$\mathit{Cl}_{0,3}$ algebra the determinant $\operatorname{Det}\mathsf{A}=0$
means $\mathsf{A}=0$. This property makes the exponential/logarithm analysis
in this algebra relatively simple. In remaining 3D algebras additional
conditions are needed to include MVs with $\operatorname{Det}\mathsf{A}=0$.,
$\operatorname{Det}(\mathbf{a}+\mathcal{A})=a_{+}^{2}a_{-}^{2}\neq 0$. If
either of scalars is zero then we have a special case. This situation is met
in rare cases, for instance222Minus sign in $a_{2}=-a_{13}$ comes from a
strictly increasing order of numbers in the basis element indices, due to the
so-called inverse degree lexicographic ordering [7]., when $a_{1}=a_{23}$,
$a_{2}=-a_{13}$, $a_{3}=a_{12}$. In such and similar cases the MVs
$\mathsf{A}_{{0}_{\pm}},\mathsf{A}_{{1,2}_{\pm}}$ and
$\mathsf{A}_{{0}_{\pm}}I$ in the Theorem 3.1 must be supplemented by
conditions:

$\displaystyle\mathsf{A}_{{0}_{+}}=$
$\displaystyle\begin{cases}\log\bigl{(}a_{0}+a_{123}\bigr{)}+2\pi
c_{2_{+}}\hat{\mathcal{U}},&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})>0\\\
\log(0_{+}),&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})=0\\\
\log\bigl{(}-(a_{0}+a_{123})\bigr{)}\\\ \qquad+(\pi+2\pi
c_{2_{+}})\hat{\mathbf{u}},&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})<0,\end{cases}$
(3.8) $\displaystyle\mathsf{A}_{{0}_{-}}=$
$\displaystyle\begin{cases}\log\bigl{(}a_{0}-a_{123})+2\pi
c_{2_{-}}\hat{\mathcal{U}},&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})>0\\\
\log(0_{+}),&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})=0\\\
\log\bigl{(}-(a_{0}-a_{123})\bigr{)}\\\ \qquad+(\pi+2\pi
c_{2_{-}})\hat{\mathbf{u}},&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})<0,\end{cases}$
(3.9) $\displaystyle\mathsf{A}_{{1,2}_{+}}=$
$\displaystyle\begin{cases}\bigr{(}\frac{1}{a_{0}+a_{123}}+2\pi
c_{1_{+}}\bigr{)}\\\
\qquad\times(1+I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})>0\\\
0,&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})=0,\\\ (\pi+2\pi
c_{1_{+}})(1+I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&a_{+}=0\quad\textrm{and}\quad(a_{0}+a_{123})<0,\\\
\end{cases}$ (3.10) $\displaystyle\mathsf{A}_{{1,2}_{-}}=$
$\displaystyle\begin{cases}\bigr{(}\frac{1}{a_{0}-a_{123}}+2\pi
c_{1_{-}}\bigr{)}\\\
\qquad\times(1-I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})>0\\\
0,&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})=0\\\ (\pi+2\pi
c_{1_{-}})(1-I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&a_{-}=0\quad\textrm{and}\quad(a_{0}-a_{123})<0,\end{cases}$
(3.11)

Here $c_{1_{\pm}},c_{2_{\pm}}\in\mathbb{Z}$ are the arbitrary integers. The
conditions for $(a_{0}\pm a_{123})$ on the right-hand side take into account
the case $\operatorname{Det}(\mathbf{a}+\mathcal{A})=0$. In scalars333The
appearance of a free vector/bivector breaks the grade arrangement in the
generic terms (3.2) and (3.3). The choice, however, results in a more simple
final expression, since now it is enough to write only a single free
$\mathbf{u}$ or $\mathcal{U}$ term (see Eq. (3.1)) instead of a pair
$\mathbf{u}$ and $\mathbf{u}I$, or $\mathcal{U}$ and $\mathcal{U}I$, if we
would have chosen to move these terms to vector+bivector part by following up
a strict grade notation convention. $\mathsf{A}_{{0}_{+}}$ and
$\mathsf{A}_{{0}_{-}}$, the symbols $\hat{\mathbf{u}}$ and $\hat{\mathcal{U}}$
represent any free unit vector or bivector, respectively,
$\hat{\mathbf{u}}^{2}=\hat{\mathcal{U}}^{2}=-1$. For example, the unit vector
can be parametrized as
$\hat{\mathbf{u}}=(u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}+u_{3}\mathbf{e}_{3})/\sqrt{u_{1}^{2}+u_{2}^{2}+u_{2}^{3}}$.
It should be noted that the term $1/(a_{0}\pm a_{123})$ in Eqs (3.10) and
(3.11) represents the limit $\lim_{a_{\pm}\to 0}\arctan(a_{0}\pm
a_{123},a_{\pm})/{a_{\pm}}=1/(a_{0}\pm a_{123})$ which is valid only when
$a_{0}\pm a_{123}>0$. The notation of $\log(0_{+})$ in expressions for
$\mathsf{A}_{{0}_{+}}$ and $\mathsf{A}_{{0}_{-}}$ is explained in Example 6.

Interpretation of special conditions (3.8)-(3.11) in terms of the MV
determinant [13, 14, 15] becomes more evident if one remembers that the
determinant of MV $\mathsf{A}$ in $\mathit{Cl}_{0,3}$ can be expressed in a
form
$\operatorname{Det}(\mathsf{A})=\bigl{(}a_{-}^{2}+(a_{0}-a_{123})^{2}\bigr{)}\bigl{(}a_{+}^{2}+(a_{0}+a_{123})^{2}\bigr{)}$,
whereas the condition $a_{\pm}=0$ is equivalent to
$\operatorname{Det}(\mathsf{A}_{{12}_{\pm}})=\operatorname{Det}(\mathbf{a}+\mathcal{A})=a_{+}^{2}a_{-}^{2}$.
All special cases therefore occur if
$\operatorname{Det}(\mathbf{a}+\mathcal{A})=0$ and the condition are described
by $a_{0}\pm a_{123}\lesseqqgtr 0$. In conclusion, the symbolic expression for
logarithm, has three special pieces (branches) $a_{0}\pm a_{123}\lesseqqgtr 0$
provided the condition $a_{\pm}=0$ is satisfied and the generic piece is
characterized by $a_{\pm}\neq 0$.

###  3.3.  Multivaluedness and free multivector

To include multivaluedness in GA logarithm we introduce a free multivector
$\mathsf{F}$ by the following defining equation [4]

$\mathrm{e}^{\log(\mathsf{A})+\mathsf{F}}=\mathrm{e}^{\log(\mathsf{A})}\mathrm{e}^{\mathsf{F}}=\mathrm{e}^{\log(\mathsf{A})},$
(3.12)

which implies two conditions the MV $\mathsf{F}$ must satisfy: the commutator
$[\log(\mathsf{A}),\mathsf{F}]=0$ and $\mathrm{e}^{\mathsf{F}}=1$. As we shall
see, for remaining $n=3$ algebras the free MV $\mathsf{F}$ will play a similar
role. One can check that the expression

$\displaystyle\mathsf{F}=$ $\displaystyle\frac{\pi
c_{1_{+}}}{a_{+}}(1+I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}+\frac{\pi
c_{1_{-}}}{a_{-}}(1-I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}$ (3.13)

satisfies $\mathrm{e}^{\mathsf{F}}=1$, and that for a generic MV $\mathsf{A}$,
Eq. 3.1, the free term (3.13) commutes with $\log(\mathsf{A})$, i.e.
$[\log(\mathsf{A}),\mathsf{F}]=0$. The integer constants
$c_{1_{+}},c_{1_{-}}\in\mathbb{Z}$ in Eq. (3.13) add two free (discrete)
parameters that may be used to shift the coefficients of vector and bivector
in $\log(\mathsf{A})$ by some multiple of $\pi$. The sum
$(\mathbf{a}+\mathcal{A})$ in (3.13) constitute vector+bivector part444In 3D
algebras, the scalar and pseudoscalar belong to algebra center and as a result
they commute with all elements. of the original MV $\mathsf{A}$, therefore
$(\mathbf{a}+\mathcal{A})$ automatically commutes with $\mathsf{A}$. As a
result only discrete free coefficients are possible in the logarithm generic
formula. In special cases (see Eqs (A.5a) and (A.5b) in the Appendix A) the
free MV $\mathsf{F}$ may also contain arbitrary unit vector $\hat{\mathbf{u}}$
and/or unit bivector $\hat{}\mathcal{U}$. In such cases one can include two
additional continuous parameters interpreted as directions of
$\hat{\mathbf{u}}$ or $\hat{}\mathcal{U}$.

Since $\arctan(x,y)$ has been defined in the range $(-\pi,\pi]$ (usually
called the principal value or the main branch, Fig. 1), we can add to it any
multiple of $2\pi$. Therefore, the plus/minus instances of $\arctan(a_{0}\pm
a_{123},a_{\pm})$ (see Eqs (A.5a) and (A.5b)) were replaced by more general
expressions $\arctan(a_{0}+a_{123},a_{+})+2\pi c_{1_{+}}$ and
$\arctan(a_{0}-a_{123},a_{-})+2\pi c_{1_{-}}$ in Eqs (3.4) and (3.5),
respectively, which takes into account the multivaluedness of the argument.
This explains the rationale behind the construction of the free MVs for GA
logarithm.

In [16] the notion of principal logarithm (also called the principal value of
logarithm) in case of matrices was introduced. In [1] it was suggested that
the ”logarithm principal value in GA can be defined as the MV
$\mathsf{M}=\log\mathsf{Y}$ with the smallest norm”. Formulas (3.2)-(3.5) and
(3.8)-(3.11) might suggest that we could obtain the principal logarithm values
after equating discrete free constants $c_{1_{\pm}},c_{2_{\pm}}$ to zero.
Unfortunately, extensive numerical checks revealed that this is not always the
case.

Example 2. Logarithm of simple MV in $\mathit{Cl}_{0,3}$. For MV
$\mathsf{A}=\tfrac{9}{10}-\tfrac{1}{3}\mathbf{e}_{3}$ in the Example 1, Eqs
(3.6) and (3.7) give $a_{+}=a_{-}=1/3$. The MVs in (3.1) then are
$\mathsf{A}_{0+}=\mathsf{A}_{0-}=-\log(10/9)+\pi\mathbf{e}_{3}$,
$\mathsf{A}_{1,2\pm}=\frac{\pi}{3}(-\mathbf{e}_{3}\pm\mathbf{e}_{12})$. The
logarithm calculated by exact formula (3.1) is
$\log(\tfrac{9}{10}-\tfrac{1}{3}\mathbf{e}_{3})=-\tfrac{1}{2}\log\frac{900}{829}-\arctan\left(\frac{10}{27}\right)\mathbf{e}_{3}\approx-0.0410873-0.354706\mathbf{e}_{3}$
which coincides with result of Example 1. Now, let’s calculate GA logarithm of
$\mathsf{A}^{\prime}=-\tfrac{9}{10}-\tfrac{1}{3}\mathbf{e}_{3}$ that diverges
when the series (2.5) is used. With exact formulas (3.2)–(3.7) we find:
$a_{+}=a_{-}=\frac{1}{3}$,
$\mathsf{A}_{0+}=\mathsf{A}_{0-}=-\frac{1}{2}\log(900/829)$,
$\mathsf{A}_{1,2+}=\big{(}\pi-\arctan(10/27)\big{)}(\mathbf{e}_{12}-\mathbf{e}_{3})$,
$\mathsf{A}_{1,2-}=\big{(}\arctan(10/27)-\pi\big{)}(\mathbf{e}_{12}+\mathbf{e}_{3})$.
Then Eq. (3.1) gives
$\log(\mathsf{A}^{\prime})=-\frac{1}{2}\log(\frac{900}{829})+(\arctan(\frac{10}{27})-\pi)\mathbf{e}_{3}\approx-0.0410873-2.78689\mathbf{e}_{3}$.
Exponentiation of the obtained logarithm gives initial MV,
$\exp\big{(}\log(\mathsf{A}^{\prime})\big{)}=\mathsf{A}^{\prime}$. The result
also can be checked by complex logarithm, because the initial MV consist of
scalar and basis vector $\mathbf{e}_{3}^{2}=-1$ only.

Example 3. Logarithm of generic MV in $\mathit{Cl}_{0,3}$. Let’s compute the
logarithm of
$\mathsf{A}=-8-6\mathbf{e}_{2}-9\mathbf{e}_{3}+5\mathbf{e}_{12}-5\mathbf{e}_{13}+6\mathbf{e}_{23}-4\mathbf{e}_{123}$.
Then, $a_{-}^{2}=53$ and $a_{+}^{2}=353$. The Eqs (3.2)-(3.5) give
$\mathsf{A}_{{0}_{+}}=\tfrac{1}{2}\log(497)$,
$\mathsf{A}_{{0}_{-}}=\tfrac{1}{2}\log(69)$,
$\mathsf{A}_{{1,2}_{+}}=(353)^{-1/2}\bigl{(}\pi-\arctan\bigl{(}\frac{\sqrt{353}}{12}\bigr{)}+2\pi
c_{1_{+}}\bigr{)}(1+I)\bigl{(}-6\mathbf{e}_{2}-9\mathbf{e}_{3}+5\mathbf{e}_{12}-5\mathbf{e}_{13}+6\mathbf{e}_{23}\bigr{)}$
and
$\mathsf{A}_{{1,2}_{-}}=(53)^{-1/2}\bigl{(}\pi-\arctan\left(\frac{\sqrt{53}}{4}\right)+2\pi
c_{1_{-}}\bigr{)}(1-I)(-6\mathbf{e}_{2}-9\mathbf{e}_{3}+5\mathbf{e}_{12}-5\mathbf{e}_{13}+6\mathbf{e}_{23})$,
where the free term $\mathsf{F}$, Eq. (3.13), has been included via
$c_{1_{\pm}}$. The logarithm is the sum of all above listed MVs:
$\log(\mathsf{A})=\frac{1}{2}\bigl{(}\mathsf{A}_{{0}_{+}}+\mathsf{A}_{{0}_{-}}+\mathsf{A}_{{1,2}_{+}}+\mathsf{A}_{{1,2}_{-}}+(\mathsf{A}_{{0}_{+}}-\mathsf{A}_{{0}_{-}})I\bigr{)}$.
Using the exponential [7] one can check that the numerical logarithm
$\log(\mathsf{A})$ indeed yields the initial MV $\mathsf{A}$ for arbitrary
integer constants $c_{1_{\pm}}$.

Example 4. Logarithm of MV when $a_{+}=0$ and $a_{0}+a_{123}>0$. The MV that
satisfies these conditions is
$\mathsf{A}=1+(3\mathbf{e}_{1}-2\mathbf{e}_{2}+\mathbf{e}_{3})+(\mathbf{e}_{12}+2\mathbf{e}_{13}+3\mathbf{e}_{23})+7\mathbf{e}_{123}=1+\mathbf{a}+\mathcal{A}+7\mathbf{e}_{123}$.
Equation (3.6) gives $a_{-}=\sqrt{56}=2\sqrt{14}$ and $a_{0}-a_{123}=-6$. Eqs
(3.8), (3.10) give $\mathsf{A}_{{0}_{+}}=\log 8+2\pi
c_{2_{+}}\hat{\mathcal{U}}$, $\mathsf{A}_{{1,2}_{+}}=\bigl{(}\frac{1}{8}+2\pi
c_{1_{+}}\bigr{)}(1+I)(\mathbf{a}+\mathbf{e}_{3}+\mathcal{A})=0$. Then from
(3.3) and (3.5) we have $\mathsf{A}_{{0}_{-}}=\tfrac{1}{2}\log 92$ and
$\mathsf{A}_{{1,2}_{-}}=\frac{\pi-\arctan\bigl{(}\frac{\sqrt{14}}{3}\bigr{)}+2\pi
c_{1_{-}}}{2\sqrt{14}}(1-I)(\mathbf{a}+\mathcal{A})$. Finally, from Eq. (3.1)
$\log(\mathsf{A})=\frac{1}{28}\Bigl{(}7\bigl{(}\log
5888-\log\frac{23}{16}\mathbf{e}_{123}\bigr{)}+\sqrt{14}\bigl{(}(2c_{1_{-}}+1)\pi-\arctan\frac{\sqrt{14}}{3}\bigr{)}(\mathbf{a}+\mathcal{A})\Bigr{)}+(1+I)\pi
c_{2_{+}}\hat{\mathcal{U}}$. After exponentiation of $\mathsf{A}$ the
constants $c_{1_{-}}$ and $c_{2_{+}}$ and bivector $\hat{\mathcal{U}}$
simplify out.

Example 5. Logarithm of MV when $a_{-}=0$ and $a_{0}-a_{123}<0$. These
conditions are satisfied by
$\mathsf{A}=1+(-3\mathbf{e}_{1}+2\mathbf{e}_{2}-\mathbf{e}_{3})+(\mathbf{e}_{12}+2\mathbf{e}_{13}+3\mathbf{e}_{23})+7\mathbf{e}_{123}=1+\mathbf{a}+\mathcal{A}+7\mathbf{e}_{123}$.
We have $a_{+}^{2}=56$, $a_{0}+a_{123}=9$ and $a_{0}-a_{123}=-6$. Then, Eq.
(3.9) gives $\mathsf{A}_{{0}_{-}}=\log 6+(\pi+2\pi
c_{2_{-}})\hat{\mathbf{u}}$. The Eqs (3.2) and (3.4) give
$\mathsf{A}_{{0}_{+}}=\frac{1}{2}\log 120$,
$\mathsf{A}_{{1,2}_{+}}=\frac{1}{2\sqrt{14}}\bigl{(}\arctan\bigl{(}\tfrac{1}{2}\sqrt{7/2}\bigr{)}+2\pi
c_{1_{+}}\bigr{)}(1+I)(\mathbf{a}+\mathcal{A})$, and Eq. (3.11)
$\mathsf{A}_{{1,2}_{-}}=(\pi+2\pi c_{1_{-}})(1-I)(\mathbf{a}+\mathcal{A})=0$.
Finally,
$\log(\mathsf{A})=\frac{1}{2}\bigl{(}\mathsf{A}_{{0}_{+}}+\mathsf{A}_{{0}_{-}}+\mathsf{A}_{{1,2}_{+}}+(\mathsf{A}_{{0}_{+}}-\mathsf{A}_{{0}_{-}})I\bigr{)}$.

Example 6. Logarithm with infinite subparts: The case $a_{+}=0$ and
$a_{0}+a_{123}=0$. The example exhibits unusual and the most interesting case.
In $\mathit{Cl}_{0,3}$, let’s compute GA logarithm of
$\mathsf{A}=1+(-2\mathbf{e}_{1}-3\mathbf{e}_{2}+5\mathbf{e}_{3})+(5\mathbf{e}_{12}+3\mathbf{e}_{13}-2\mathbf{e}_{23})-\mathbf{e}_{123}=1+\mathbf{a}+\mathcal{A}-\mathbf{e}_{123}$.
The remaining scalar is $a_{-}=2\sqrt{38}$, $(a_{0}-a_{123})=2$. Then, Eq.
(3.8) gives $\mathsf{A}_{{0}_{+}}=\log(0_{+})$; Eq. (3.3) gives
$\mathsf{A}_{{0}_{-}}=\tfrac{1}{2}\log 156$; Eq. (3.10) gives
$\mathsf{A}_{{1,2}_{+}}=0$; Eq. (3.5) gives
$\mathsf{A}_{{1,2}_{-}}=\frac{\arctan\left(\sqrt{38}\right)+2\pi
c_{1_{-}}}{\sqrt{38}}(\mathbf{a}+\mathcal{A})$. Finally, the logarithm of
$\mathsf{A}$ is

$\begin{split}\log(\mathsf{A})=&\frac{\arctan\left(\sqrt{38}\right)+2\pi
c_{1_{-}}}{2\sqrt{38}}\left(\mathbf{a}+\mathcal{A}\right)\\\
&+\tfrac{1}{2}\big{(}\log(0_{+})\left(1+\mathbf{e}_{123}\right)+\tfrac{1}{2}\log(156)\left(1-\mathbf{e}_{123}\right)\big{)}\,.\end{split}$
(3.14)

Note the factor $\log(0_{+})$ in front of $\left(1+\mathbf{e}_{123}\right)$.
If logarithm in this form is inserted into coordinate-free exponential [8] we
will get

$\left(\tfrac{1}{2}\mathrm{e}^{\log(0_{+})}+1\right)+\mathbf{a}+\mathcal{A}+\left(\tfrac{1}{2}\mathrm{e}^{\log(0_{+})}-1\right)\mathbf{e}_{123}\,,$
(3.15)

which coincides with the initial MV if we assume that555The statement can be
made strict by considering the limit $\lim_{x\to 0_{+}}\exp(\log(x))=0$, where
$x\to 0_{+}$ indicates that the limit is taken keeping $x$ positive, i.e.”from
above”. $\log(0_{+})=-\infty$.

###  3.4. GA Logarithm of blades and their combinations in $\mathit{Cl}_{0,3}$

In this subsection, the logarithms for individual blades and their
combinations that follow from generic logarithm (Theorem 3.1), and may be
useful in practice are collected. The norms listed below are positive scalars.
Vector norm:
$\lvert\mathbf{a}\rvert=\sqrt{\mathbf{a}\wideparen{\mathbf{a}}}=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$
.
Paravector norm: $\lvert
a_{0}+\mathbf{a}\rvert=\lvert\mathsf{A}_{0,1}\rvert=\bigl{(}\mathsf{A}_{0,1}\wideparen{\mathsf{A}}_{0,1}\bigr{)}^{\frac{1}{2}}=\sqrt{a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$
.
Bivector norm:
$\lvert\mathcal{A}\rvert=\bigl{(}\mathcal{A}\widetilde{\mathcal{A}}\bigr{)}^{\frac{1}{2}}=\sqrt{a_{12}^{2}+a_{13}^{2}+a_{23}^{2}}$
.
Rotor norm: $\lvert
a_{0}+\mathcal{A}\rvert=\lvert\mathsf{A}_{0,2}\rvert=\bigl{(}\mathsf{A}_{0,2}\widetilde{\mathsf{A}}_{0,2}\bigr{)}^{\frac{1}{2}}=\sqrt{a_{0}^{2}+a_{12}^{2}+a_{13}^{2}+a_{23}^{2}}$
.

Logarithms of blades and their combinations.

Logarithm of vector
$\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}$,
$c_{i}\in\mathbb{Z}$,

$\displaystyle\log(\mathbf{a})=$
$\displaystyle\frac{1}{2}\log(\lvert\mathbf{a}\rvert^{2})+\pi\frac{\mathbf{a}}{\lvert\mathbf{a}\rvert}\bigl{(}\frac{1}{2}+c_{1}(1+I)+c_{2}(1-I)\bigr{)},$
$\displaystyle\lvert\mathbf{a}\rvert^{2}\neq 0.$ (3.16)

Logarithm of paravector $\mathsf{A}_{0,1}=a_{0}+\mathbf{a}$;
$c_{i}\in\mathbb{Z}$ and $\hat{\mathbf{u}}^{2}=-1$,
$\hat{\mathcal{U}}^{2}=-1$.

$\displaystyle\log\mathsf{A}_{0,1}=$ $\displaystyle\begin{aligned}
&\frac{1}{2}\log(\lvert\mathsf{A}_{0,1}\rvert^{2})+\frac{\mathbf{a}}{\lvert\mathbf{a}\rvert}\bigl{(}\arctan(a_{0},\lvert\mathbf{a}\rvert)\\\
&\qquad\qquad+\pi(c_{1}(1+I)+c_{2}(1-I))\bigr{)},\end{aligned}$
$\displaystyle\qquad\lvert\mathbf{a}\rvert\neq 0.$ (3.17)

Logarithm of bivector
$\mathcal{A}=a_{12}\mathbf{e}_{12}+a_{13}\mathbf{e}_{13}+a_{23}\mathbf{e}_{23}$,
$c_{i}\in\mathbb{Z}$,

$\displaystyle\log(\mathcal{A})=$
$\displaystyle\frac{1}{2}\log(\lvert\mathcal{A}\rvert^{2})+\pi\frac{\mathcal{A}}{\lvert\mathcal{A}\rvert}\bigl{(}\frac{1}{2}+c_{1}(1+I)+c_{2}(1-I)\bigr{)},$
$\displaystyle\lvert\mathcal{A}\rvert^{2}\neq 0.$ (3.18)

Logarithm of parabivector and rotor $\mathsf{A}_{0,2}=a_{0}+\mathcal{A}$,

$\displaystyle\log\mathsf{A}_{0,2}=$ $\displaystyle\begin{aligned}
&\frac{1}{2}\log(\lvert\mathsf{A}_{0,2}\rvert^{2})+\frac{\mathcal{A}}{\lvert\mathcal{A}\rvert}\bigl{(}\arctan(a_{0},\lvert\mathcal{A}\rvert)\\\
&\qquad\qquad+\pi(c_{1}(1+I)+c_{2}(1-I))\bigr{)},\end{aligned}$
$\displaystyle\qquad\lvert\mathcal{A}\rvert\neq 0.$ (3.19)

Logarithm of center $\mathsf{A}_{0,3}=a_{0}+a_{123}I$,

$\displaystyle\log\mathsf{A}_{0,3}=$
$\displaystyle\begin{cases}\begin{aligned}
&\bigl{(}\tfrac{1}{2}\log(a_{0}-a_{123})+\pi
c_{1}\hat{\mathcal{U}}_{1}\bigr{)}\bigl{(}1-I\bigr{)}\\\
&+\bigl{(}\tfrac{1}{2}\log(a_{0}+a_{123})+\pi
c_{2}\hat{\mathcal{U}}_{2}\bigr{)}\bigl{(}1+I\bigr{)},\end{aligned}&\begin{aligned}
(a_{0}-a_{123})&>0\textrm{ and }\\\ (a_{0}+a_{123})&>0\\\
\end{aligned}\\\\[17.0pt] \begin{aligned}
&\bigl{(}\tfrac{1}{2}\log(a_{0}-a_{123})+\pi
c_{1}\hat{\mathcal{U}}_{1}\bigr{)}\bigl{(}1-I\bigr{)}\\\
&+\bigl{(}\tfrac{1}{2}\log(-a_{0}-a_{123})+\pi(c_{2}+\tfrac{1}{2})\hat{\mathbf{u}}_{2}\bigr{)}\bigl{(}1+I\bigr{)},\end{aligned}&\begin{aligned}
(a_{0}-a_{123})&>0\textrm{ and }\\\ (a_{0}+a_{123})&<0\\\
\end{aligned}\\\\[17.0pt] \begin{aligned}
&\bigl{(}\tfrac{1}{2}\log(-a_{0}+a_{123})+\pi(c_{1}+\tfrac{1}{2})\hat{\mathbf{u}}_{1}\bigr{)}\bigl{(}1-I\bigr{)}\\\
&+\bigl{(}\tfrac{1}{2}\log(a_{0}+a_{123})+\pi
c_{2}\hat{\mathcal{U}}_{2}\bigr{)}\bigl{(}1+I\bigr{)},\end{aligned}&\begin{aligned}
(a_{0}-a_{123})&<0\textrm{ and }\\\ (a_{0}+a_{123})&>0\\\
\end{aligned}\\\\[17.0pt] \begin{aligned}
&\bigl{(}\tfrac{1}{2}\log(-a_{0}+a_{123})+\pi(c_{1}+\tfrac{1}{2})\hat{\mathbf{u}}_{1}\bigr{)}\bigl{(}1-I\bigr{)}\\\
&+\bigl{(}\tfrac{1}{2}\log(-a_{0}-a_{123})+\pi(c_{2}+\tfrac{1}{2})\hat{\mathbf{u}}_{2}\bigr{)}\bigl{(}1+I\bigr{)},\end{aligned}&\begin{aligned}
(a_{0}-a_{123})&<0\textrm{ and }\\\ (a_{0}+a_{123})&<0\\\
\end{aligned}\end{cases}$ (3.20)

where $\hat{\mathbf{u}}_{i}$ and $\hat{\mathcal{U}}_{j}$ are arbitrary non-
commuting unit vector and bivector, respectively. If $(a_{0}-a_{123})=0$ or
$(a_{0}+a_{123})=0$ some of subparts give $\log(0_{+})$.

##  4\. MV logarithms in $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$

$\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$ algebras are isomorphic. Their
multiplication tables coincide, for example, after the following exchange of
basis elements:

$\begin{split}\mathit{Cl}_{3,0}\quad&\\{1,\quad\mathbf{e}_{1},\quad\mathbf{e}_{2},\quad\mathbf{e}_{3},\
\ \mathbf{e}_{12},\ \ \mathbf{e}_{13},\ \mathbf{e}_{23},\
\mathbf{e}_{123}\\}\downarrow\\\ \mathit{Cl}_{1,2}\quad&\\{1,\
-\mathbf{e}_{1},-\mathbf{e}_{12},-\mathbf{e}_{13},-\mathbf{e}_{2},-\mathbf{e}_{3},\
\mathbf{e}_{23},-\mathbf{e}_{123}\\}.\end{split}$

To find formulas for logarithm in coordinate form the same inverse solution
method was used as described in the Appendix A for $\mathit{Cl}_{0,3}$
algebra. The logarithm in $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$ exists
for all MVs except of nonzero MVs of the form
$\mathsf{A}_{1,2}=\mathbf{a}+\mathcal{A}$ that satisfy the condition
$\operatorname{Det}(\mathsf{A}_{1,2})=(a_{+}^{2}+a_{-}^{2})^{2}=0$, i.e., for
MVs that are the sums of vector and bivector and the determinant are equal to
zero. These restrictions are the same as those for GA square root to exist
(see [10] and Example 3 herein in case $s=S=0$).

###  4.1. Logarithm formula for generic MV

######  Theorem 4.1 (Logarithm of multivector in $\mathit{Cl}_{3,0}$ and
$\mathit{Cl}_{1,2}$).

The logarithm of generic MV $\mathsf{A}=a_{0}+\mathbf{a}+\mathcal{A}+a_{123}I$
is another MV

$\log(\mathsf{A})=\mathsf{A}_{{0}}+\mathsf{A}_{{1,2}_{\log}}+\mathsf{A}_{{1,2}_{\arctan}}+\mathsf{A}_{I},$
(4.1)

where

$\displaystyle\mathsf{A}_{{0}}$ $\displaystyle=\frac{1}{2}\bigl{(}\log
k_{+}+\log k_{-}\bigr{)},$ $\displaystyle a_{+}^{2}+a_{-}^{2}\neq
0\allowdisplaybreaks$ (4.2) $\displaystyle\mathsf{A}_{{1,2}_{{\log}}}$
$\displaystyle=\frac{1}{2}\frac{a_{+}-a_{-}I}{a_{-}^{2}+a_{+}^{2}}\bigl{(}\log
k_{+}-\log k_{-}\bigr{)}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},$
$\displaystyle a_{+}^{2}+a_{-}^{2}\neq 0$ (4.3)
$\mathsf{A}_{{1,2}_{\arctan}}=I\frac{a_{+}-a_{-}I}{a_{-}^{2}+a_{+}^{2}}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}\Bigl{(}\frac{1}{2}\arctan\bigl{(}-(a_{+}^{2}-a_{0}^{2})-(a_{-}^{2}-a_{123}^{2}),\\\\[-1.0pt]
\hfill(a_{+}-a_{0})(a_{-}+a_{123})-(a_{+}+a_{0})(a_{-}-a_{123})\bigr{)}+2\pi
c_{1}\Bigr{)},\\\ \hfill\textrm{when}\quad(a_{+}^{2}+a_{-}^{2}\neq
0)\quad\textrm{and}\quad(k_{-}k_{+}\neq 0),$ (4.4)
$\mathsf{A}_{I}=I\arctan\bigl{(}(a_{+}+a_{0})k_{-}-(a_{+}-a_{0})k_{+},(a_{-}+a_{123})k_{-}-(a_{-}-a_{123})k_{+}\bigr{)}\\\
\qquad+2\pi c_{2}I,\qquad\hfil\textrm{when}\quad(a_{+}^{2}+a_{-}^{2}\neq
0)\quad\textrm{and either}\\\ (a_{+}+a_{0})k_{-}-(a_{+}-a_{0})k_{+}\neq
0\quad\textrm{or}\quad(a_{-}+a_{123})k_{-}-(a_{-}-a_{123})k_{+}\neq 0$ (4.5)

where scalar coefficients are

$\displaystyle k_{-}^{2}$
$\displaystyle=(a_{+}-a_{0})^{2}+(a_{-}-a_{123})^{2},\qquad
k_{+}^{2}=(a_{+}+a_{0})^{2}+(a_{-}+a_{123})^{2},$ (4.6)

and

$\displaystyle
a_{-}=\frac{-2I\mathbf{a}\mathbin{\wedge}\mathcal{A}}{\sqrt{2}\sqrt{\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A}+\sqrt{(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})^{2}-4(\mathbf{a}\mathbin{\wedge}\mathcal{A})^{2}}}},$
$\displaystyle
a_{+}=\frac{\sqrt{\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A}+\sqrt{(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})^{2}-4(\mathbf{a}\mathbin{\wedge}\mathcal{A})^{2}}}}{\sqrt{2}}\,$
(4.7) $\displaystyle\quad\textrm{for
}\mathbf{a}\mathbin{\wedge}\mathcal{A}\neq
0,\quad\textrm{and}\allowdisplaybreaks$
$\displaystyle\begin{cases}a_{+}=\sqrt{\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A}},\quad
a_{-}=0,&\textrm{if}\quad\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A}\geq
0\\\ a_{+}=0,\quad
a_{-}=\sqrt{-(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})},&\textrm{if}\quad\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A}<0,\end{cases}$
$\displaystyle\qquad\textrm{when}\quad\mathbf{a}\mathbin{\wedge}\mathcal{A}=0.$

The constants $c_{1},c_{2}$ are arbitrary integers.

Proof. It is enough to check that after substitution of $\log{\mathsf{A}}$
expressions into exponential formula presented in [7] one gets the initial MV
$\mathsf{A}$. The factor $\frac{a_{+}-a_{-}I}{a_{-}^{2}+a_{+}^{2}}$ in the
above formulas alternatively may be written as $(a_{+}+a_{-}I)^{-1}$.

###  4.2. Special cases

When the conditions listed in Eqs (4.2)-(4.5) are not satisfied, we have
special cases. In particular, the condition $k_{\pm}=0$ means that the MV
determinant is zero, $\operatorname{Det}(\mathsf{A})=k_{-}^{2}k_{+}^{2}=0$.
Similarly, the condition $a_{+}^{2}+a_{-}^{2}=0$ implies that determinant of
vector+bivector part vanishes,
$\operatorname{Det}(\mathsf{A}_{1,2})=(a_{+}^{2}+a_{-}^{2})^{2}=0$. The
specific relations $(a_{+}+a_{0})k_{-}-(a_{+}-a_{0})k_{+}\neq 0$ and
$(a_{-}+a_{123})k_{-}-(a_{-}-a_{123})k_{+}\neq 0$ in Eq. (4.5) as well as the
relation $k_{-}k_{+}\neq 0$ in Eq. (4.4) ensure that both arguments of
$\arctan(x,y)$ do not nullify simultaneously.

When the generic formula is not applicable the expressions for
$\mathsf{A}_{{0}},\mathsf{A}_{{1,2}_{{\log}}},\mathsf{A}_{{1,2}_{\arctan}}$
and $\mathsf{A}_{I}$ in Theorem 4.1 must be supplemented by following formulas

$\displaystyle\begin{split}\mathsf{A}_{{0}}{}&=\begin{cases}\frac{1}{2}\log\bigl{(}a_{0}^{2}+a_{123}^{2}\bigr{)},&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=&0)\land(a_{0}^{2}+a_{123}^{2}\neq 0),\end{aligned}\\\
\varnothing,&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}=0)\end{aligned}\end{cases}\end{split}$
(4.8)
$\displaystyle\begin{split}\mathsf{A}_{{1,2}_{{\log}}}{}&=\begin{cases}0,&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}\neq 0),\end{aligned}\\\
\varnothing,&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}=0)\end{aligned}\end{cases}\end{split}$
(4.9)
$\displaystyle\begin{split}\mathsf{A}_{{1,2}_{\arctan}}{}&=\begin{cases}\pi(\frac{1}{2}+2c_{1})I\frac{a_{+}-a_{-}I}{a_{-}^{2}+a_{+}^{2}}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}\neq 0)\land(k_{-}k_{+}=0),\end{aligned}\\\
\frac{a_{0}-a_{123}I}{a_{0}^{2}+a_{123}^{2}}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}+\hat{\mathcal{F}},&\begin{aligned}
&(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}\neq 0),\end{aligned}\\\
\varnothing,&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=0)&\land(a_{0}^{2}+a_{123}^{2}=0)\end{aligned}\end{cases}\end{split}$
(4.10)
$\displaystyle\begin{split}\mathsf{A}_{I}{}&=\begin{cases}I\bigl{(}\arctan(-a_{-},a_{+})+2\pi
c_{2}\bigr{)},&\begin{aligned} &(a_{+}^{2}+a_{-}^{2}\neq 0)\\\
&\land((a_{+}+a_{0})k_{-}-(a_{+}-a_{0})k_{+}=0)\\\
&\land((a_{-}+a_{123})k_{-}-(a_{-}-a_{123})k_{+}=0),\end{aligned}\\\
I\bigl{(}\arctan(a_{0},a_{123})+2\pi c_{2}\bigr{)},&\begin{aligned}
&(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}\neq 0),\end{aligned}\\\
\varnothing,&\begin{aligned}
(a_{+}^{2}+a_{-}^{2}=0)&\land(a_{0}^{2}+a_{123}^{2}=0)\end{aligned}\end{cases}\end{split}$
(4.11)

Here the symbols $\land$ and $\lor$ in the conditions represent logical
conjunction and disjunction, respectively.
$\hat{\mathcal{F}}=\begin{cases}2\pi
c_{1}\hat{\mathcal{U}},&\mathrm{if}\quad\mathbf{a}+\mathcal{A}=0\\\
0,&\mathrm{if}\quad\mathbf{a}+\mathcal{A}\neq 0\end{cases}$, where the free
unit bivector must satisfy $\hat{\mathcal{U}}^{2}=-1$. After exponentiation it
gives $\exp(\hat{\mathcal{U}})=1$ and represents continuous degree of freedom
(a direction) in (4.10) and (4.11), and can be parameterized as

$\hat{\mathcal{U}}=\begin{cases}\frac{d_{12}\mathbf{e}_{12}+d_{13}\mathbf{e}_{13}+d_{23}\mathbf{e}_{23}}{\sqrt{d_{12}^{2}+d_{13}^{2}+d_{23}^{2}}},&\mathrm{for}\quad\mathit{Cl}_{3,0}\\\
\frac{d_{12}\mathbf{e}_{12}+d_{13}\mathbf{e}_{13}+d_{23}\mathbf{e}_{23}}{\sqrt{-d_{12}^{2}-d_{13}^{2}+d_{23}^{2}}},&\mathrm{for}\quad\mathit{Cl}_{1,2},\
\mathrm{when}\ -d_{12}^{2}-d_{13}^{2}+d_{23}^{2}>0\end{cases}$ (4.12)

The cases $k_{\pm}=0$ that represent MV with a vanishing determinant,
$\operatorname{Det}(\mathsf{A})=k_{-}^{2}k_{+}^{2}=0$, yield MVs with infinite
coefficients (see Example 9 for details).

###  4.3. Multiveluedness and free multivector

In Eqs (4.4) and (4.5) we may add any multiple of $2\pi$ to both arc tangent
functions, i.e. $\arctan(y_{1},y_{2})\to\arctan(y_{1},y_{2})+2\pi c_{i}$.
After collecting terms in front of free coefficients
$c_{1},c_{2}\in\mathbb{Z}$, we obtain a free MV $\mathsf{F}$ that satisfies
$\exp(\mathsf{F})=1$,

$\displaystyle\mathsf{F}=$ $\displaystyle\frac{2\pi
c_{1}}{\bigl{(}a_{-}^{2}+a_{+}^{2}\bigr{)}}\big{(}a_{-}(\mathbf{a}+\mathcal{A})+a_{+}(\mathbf{a}+\mathcal{A})I\big{)}+2\pi
c_{2}I\,,$ (4.13)

where $a_{\pm}$ are given by Eq. (4.7).

Example 7. Logarithm of generic MV in $\mathit{Cl}_{3,0}$. Let us take simple
but representative MV:
$\mathsf{A}=-2+\mathbf{e}_{1}+\mathbf{e}_{23}-3\mathbf{e}_{123}$. From Eqs
(4.6) and (4.7) we have $k_{+}^{2}=5$, $k_{-}^{2}=25$ and $a_{+}=a_{-}=1$.
Then (4.2) and (4.3) yield $\mathsf{A}_{{0}}=\frac{3\log 5}{4}$ and
$\mathsf{A}_{{1,2}_{\log}}=-\frac{\log
5}{8}\bigl{(}\mathbf{e}_{1}+\mathbf{e}_{23}\bigr{)}(1-I)$. Next, the Eqs (4.4)
and (4.5) give
$\mathsf{A}_{{1,2}_{\arctan}}=-\frac{1}{4}\bigl{(}-\arctan\frac{2}{11}+4\pi
c_{2}\bigr{)}\bigl{(}\mathbf{e}_{1}+\mathbf{e}_{23}\bigr{)}(1+I)$ and
$\mathsf{A}_{I}=\Bigl{(}-\pi+\arctan\frac{-10-4\sqrt{5}}{-5-3\sqrt{5}}+2\pi
c_{1}\Bigr{)}\mathbf{e}_{123}$. Finally, after summation of all terms in (4.1)
we obtain $\log(\mathsf{A})=\frac{\log
5}{4}\bigl{(}3-\mathbf{e}_{1}\bigr{)}+\frac{1}{2}\arctan\frac{2}{11}\mathbf{e}_{23}+\bigl{(}-\pi+\arctan(\frac{1}{2}(1+\sqrt{5})\bigr{)}\mathbf{e}_{123}+\mathsf{F}$,
where the free MV
$\mathsf{F}=2\pi\bigl{(}c_{1}\mathbf{e}_{123}-c_{2}\mathbf{e}_{23}\bigr{)}$.
The coefficients $c_{1},c_{2}\in\mathbb{Z}$ come from
$\mathsf{A}_{{1,2}_{\arctan}}$ and $\mathsf{A}_{I}$ terms, respectively.
Substitution of this result into exponential $\exp(\log(\mathsf{A}))$ returns
the initial MV.

Example 8. Logarithm of center of $\mathit{Cl}_{3,0}$.
$\mathsf{A}=1-2\mathbf{e}_{123}$. Since $\mathbf{e}_{123}^{2}=-1$ the MV is a
counterpart of complex number logarithm. Eqs (4.6) and (4.7) give
$a_{+}=a_{-}=0$ and $k_{+}^{2}=k_{-}^{2}=5$. Then, Eq. (4.8) gives
$\mathsf{A}_{{0}}=\frac{\log 5}{2}$; Eq. (4.9) gives
$\mathsf{A}_{{1,2}_{\log}}=0$; Eq. (4.10) gives
$\mathsf{A}_{{1,2}_{\arctan}}=2\pi c_{1}\hat{\mathcal{U}}$; Eq. (4.11) gives
$\mathsf{A}_{I}=\bigl{(}-\arctan 2+2\pi c_{2}\bigr{)}\mathbf{e}_{123}$. Note
that $\hat{\mathcal{U}}$ is the same free MV for both
$\mathsf{A}_{{1,2}_{\arctan}}$. After summation of terms in (4.1) the final
answer is $\log(\mathsf{A})=\frac{\log 5}{2}+\bigl{(}-\arctan 2+2\pi
c_{2}\bigr{)}\mathbf{e}_{123}+2\pi c_{1}\hat{\mathcal{U}}$. On the other hand
the complex number $1-2\,\mathrm{i}$ gives
$\log(1-2\,\mathrm{i})=\tfrac{1}{2}\log 5-\arctan 2$, which coincides with
$\mathit{Cl}_{3,0}$ algebra result if $c_{1}=c_{2}=0$.

Example 9. Logarithm of singular MV when $\operatorname{Det}(\mathsf{A})=0$.
This is the most intriguing and complicated case in $\mathit{Cl}_{3,0}$. Since
$\operatorname{Det}(\mathsf{A})=k_{-}^{2}k_{+}^{2}$ we may have either
$k_{-}^{2}=0$ or $k_{+}^{2}=0$. The case when $k_{-}^{2}=k_{+}^{2}=0$ is
trivial since it requires all MV components to vanish. Let’s analyze the case
when $k_{+}^{2}\neq 0$ and $k_{-}^{2}=0$. It is represented, for example, by
$\mathsf{A}=6+(-8\mathbf{e}_{1}-2\mathbf{e}_{3})+(-\mathbf{e}_{12}+10\mathbf{e}_{13}+10\mathbf{e}_{23})-13\mathbf{e}_{123}=6+\mathbf{a}+\mathcal{A}-13\mathbf{e}_{123}$.
From Eq. (4.7) we find $a_{+}=6$, $a_{-}=-13$ and from (4.6) $k_{+}^{2}=820$,
$k_{-}^{2}=0$. Then, Eq. (4.2) gives
$\mathsf{A}_{{0}}=\frac{1}{2}\bigl{(}\log(2\sqrt{205})+\log(0_{+})\bigr{)}$;
Eq. (4.3) gives
$\mathsf{A}_{{1,2}_{{\log}}}=\frac{1}{410}\bigl{(}\log(2\sqrt{205})-\log(0_{+})\bigr{)}\bigl{(}6+13\mathbf{e}_{123}\bigr{)}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}+6(\mathbf{a}+\mathcal{A})\bigr{)}$;
Eq. (4.10) gives
$\mathsf{A}_{{1,2}_{\arctan}}=\frac{\pi}{205}(\frac{1}{2}+2c_{1})\bigl{(}-6+13\mathbf{e}_{123}\bigr{)}\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}+6(\mathbf{a}+\mathcal{A})\bigr{)}$;
finally, Eq. (4.11) gives $\mathsf{A}_{I}=\bigl{(}\arctan(\frac{6}{13})+2\pi
c_{2}\bigr{)}\mathbf{e}_{123}$. Summing up all terms we obtain the answer

$\displaystyle\log\mathsf{A}=$
$\displaystyle\frac{1}{2}\bigl{(}\log(2\sqrt{205})+\log(0_{+})\bigr{)}+\Bigl{(}\frac{1}{410}\bigl{(}\log(2\sqrt{205})-\log(0_{+})\bigr{)}\bigl{(}6+13\mathbf{e}_{123}\bigr{)}$
$\displaystyle\quad+\frac{\pi}{205}(\frac{1}{2}+2c_{1})\bigl{(}-6+13\mathbf{e}_{123}\bigr{)}\Bigr{)}(\mathbf{a}+\mathcal{A})+\bigl{(}\arctan(\frac{6}{13})+2\pi
c_{2}\bigr{)}\mathbf{e}_{123}.$

The result can be checked after replacement of $\log(0_{+})$ by $\log(x)$ and
substitution into exponential formula (4.1) of paper [7]. After simplification
one can take the limit $\lim_{x\to 0_{+}}\exp\bigl{(}\log\mathsf{A}\bigr{)}$,
which returns the initial MV. This example demonstrates that the logarithm of
MV with specific finite coefficients may yield MV with some of coefficients in
the answer being infinite and which have to be understood as the limit
$\lim_{x\to 0_{+}}\log(x)$. The answer, nevertheless, is meaningful since the
substitution of the answer into exponential formula and computation of the
limit reproduces the initial MV.

###  4.4. Logarithms of individual blades and their combinations

Below we use different norms for individual blades of $\mathit{Cl}_{3,0}$,
since a positive scalar for vectors and bivectors is calculated differently.
In particular, for a vector we will use
$\lvert\mathbf{a}\rvert=\sqrt{\mathbf{a}\mathbf{a}}=\sqrt{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}$
, whereas for a bivector
$\lvert\mathcal{A}\rvert=\bigl{(}\mathcal{A}\widetilde{\mathcal{A}}\bigr{)}^{1/2}=\sqrt{a_{12}^{2}+a_{13}^{2}+a_{23}^{2}}$
. For a rotor $\lvert
a_{0}+\mathcal{A}\rvert=\lvert\mathsf{A}_{0,2}\rvert=\bigl{(}\mathsf{A}_{0,2}\widetilde{\mathsf{A}}_{0,2}\bigr{)}^{1/2}=\sqrt{a_{0}^{2}+a_{12}^{2}+a_{13}^{2}+a_{23}^{2}}$
, and for an element of center $\lvert
a_{0}+a_{123}I\rvert=\lvert\mathsf{A}_{0,3}\rvert=\bigl{(}\mathsf{A}_{0,3}\wideparen{\mathsf{A}}_{0,3}\bigr{)}^{1/2}=\sqrt{a_{0}^{2}+a_{123}^{2}}$
.

Logarithm of vector:
$\mathbf{a}=a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3}$,

$\displaystyle\log(\mathbf{a})=$
$\displaystyle\frac{1}{2}\log(\lvert\mathbf{a}\rvert^{2})-\pi\bigl{(}\tfrac{1}{2}+2c_{2}\bigr{)}\frac{\mathbf{a}}{\lvert\mathbf{a}\rvert}I+\pi\bigl{(}\tfrac{1}{2}+2c_{1}\bigr{)}I,$
$\displaystyle\lvert\mathbf{a}\rvert^{2}\neq 0\,.$ (4.14)

Logarithm of bivector:
$\mathcal{A}=a_{12}\mathbf{e}_{12}+a_{13}\mathbf{e}_{13}+a_{23}\mathbf{e}_{23}$,

$\displaystyle\log(\mathcal{A})=$
$\displaystyle\frac{1}{2}\log(\lvert\mathcal{A}\rvert^{2})-\pi\bigl{(}\tfrac{1}{2}+2c_{2}\bigr{)}\frac{\mathcal{A}}{\lvert\mathcal{A}\rvert}+\pi(1+2c_{1})I,$
$\displaystyle\lvert\mathcal{A}\rvert^{2}\neq 0\,.$ (4.15)

Logarithm of rotor: $\mathsf{A}_{0,2}=a_{0}+\mathcal{A}$,

$\displaystyle\log\mathsf{A}_{0,2}=$
$\displaystyle\begin{cases}\begin{aligned}
&\frac{1}{2}\log(\lvert\mathsf{A}_{0,2}\rvert^{2})+\bigl{(}\arctan\bigl{(}a_{0},0\bigr{)}+2\pi
c_{1}\bigr{)}I\\\
&\quad+\frac{\mathcal{A}}{\lvert\mathcal{A}\rvert}\Bigl{(}2\pi
c_{2}-\frac{1}{2}\arctan\bigl{(}a_{0}^{2}-\lvert\mathcal{A}\rvert^{2},-2a_{0}\lvert\mathcal{A}\rvert\bigr{)}\Bigr{)}\end{aligned},&\begin{aligned}
\lvert\mathsf{A}_{0,2}\rvert\neq 0\end{aligned}\\\ \log a_{0}+2\pi
c_{1}I,&\lvert\mathcal{A}\rvert=0\textrm{ and }a_{0}\geq 0\\\
\log(-a_{0})+2\pi(c_{1}+1)I,&\lvert\mathcal{A}\rvert=0\textrm{ and }a_{0}<0\\\
\text{see bivector formula~{}\eqref{BivForCl30}},&\lvert\mathcal{A}\rvert\neq
0\textrm{ and }a_{0}=0,\end{cases}$ (4.16)

Logarithm of center:
$\mathsf{A}_{0,3}=a_{0}+a_{123}\mathbf{e}_{123}=a_{0}+a_{123}I$,
$\lvert\mathsf{A}_{0,3}\rvert^{2}=\mathsf{A}_{0,3}\widetilde{\mathsf{A}}_{0,3}$.

$\displaystyle\log(\mathsf{A}_{0,3})=$
$\displaystyle\begin{cases}\frac{1}{2}\log(\lvert\mathsf{A}_{0,3}\rvert^{2})+2\pi
c_{2}\hat{\mathcal{U}}+\bigr{(}\arctan\bigl{(}a_{0},a_{123}\bigr{)}+4\pi
c_{1}\bigl{)}I,&\lvert\mathsf{A}_{0,3}\rvert^{2}\neq 0,\\\ \log(0_{+})+2\pi
c_{2}\hat{\mathcal{U}},&\lvert\mathsf{A}_{0,3}\rvert^{2}=0\,.\\\ \end{cases}$
(4.17)

The paravector $\mathsf{A}_{0,1}=a_{0}+\mathbf{a}$ norm $\lvert
a_{0}+\mathbf{a}\rvert^{2}\equiv\lvert\mathsf{A}_{0,1}\rvert^{2}=\mathsf{A}_{0,1}\wideparen{\mathsf{A}}_{0,1}=a_{0}^{2}-a_{1}^{2}-a_{2}^{2}-a_{3}^{2}$,
contains coefficients with opposite signs. The logarithm formula, therefore,
splits into many subcases and is impractical.

##  5\. MV logarithms in $\mathit{Cl}_{2,1}$

Of all three algebras, the logarithm of $\mathit{Cl}_{2,1}$ appeared the most
hard to recover. The logarithms in $\mathit{Cl}_{3,0}$ and $\mathit{Cl}_{1,2}$
algebras exist for almost all MVs except very small specific class of vectors
and bivectors, $\mathbf{a}+\mathcal{A}\neq 0$, with the vanishing determinant
$\operatorname{Det}{(\mathbf{a}+\mathcal{A})}=0$. In $\mathit{Cl}_{2,1}$
algebra the logarithm does not exist for a large class of MVs. In contrast, in
$\mathit{Cl}_{0,3}$ algebra the logarithm exists for all MVs.

######  Theorem 5.1.

[Logarithm of multivector in $\mathit{Cl}_{2,1}$] The logarithm of multivector
$\mathsf{A}=a_{0}+(a_{1}\mathbf{e}_{1}+a_{2}\mathbf{e}_{2}+a_{3}\mathbf{e}_{3})+(a_{12}\mathbf{e}_{12}+a_{13}\mathbf{e}_{13}+a_{23}\mathbf{e}_{23})+a_{123}I=a_{0}+\mathbf{a}+\mathcal{A}+a_{123}I$
is the MV

$\displaystyle\log(\mathsf{A})=$
$\displaystyle\begin{cases}\frac{1}{2}\bigl{(}\mathsf{A}_{{0}_{+}}+\mathsf{A}_{{0}_{-}}+\mathsf{A}_{{1,2}_{+}}+\mathsf{A}_{{1,2}_{-}}+(\mathsf{A}_{{0}_{+}}-\mathsf{A}_{{0}_{-}})I\bigr{)},&f_{\pm}\geq
0\\\ \varnothing,&f_{\pm}<0\end{cases}\allowdisplaybreaks$ (5.1) where
$\displaystyle\begin{aligned} &f_{\pm}=(a_{0}\pm
a_{123})^{2}+a_{\pm}^{2},&f_{\pm}\lesseqqgtr 0,\\\
&a_{-}^{(2)}=-(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})+2I\mathbf{a}\mathbin{\wedge}\mathcal{A},&a_{-}^{(2)}\lesseqqgtr
0,\\\
&a_{+}^{(2)}=-(\mathbf{a}\mathbin{\cdot}\mathbf{a}+\mathcal{A}\mathbin{\cdot}\mathcal{A})-2I\mathbf{a}\mathbin{\wedge}\mathcal{A},&a_{+}^{(2)}\lesseqqgtr
0,\allowdisplaybreaks\\\ \end{aligned}$ (5.2) and
$\displaystyle\mathsf{A}_{{0}_{\pm}}=$
$\displaystyle\begin{cases}\tfrac{1}{2}\log(f_{\pm}),&\begin{aligned}
(a_{\pm}^{(2)}>0)\end{aligned}\\\ \begin{aligned}
&\tfrac{1}{2}\log\Bigl{(}a_{0}\pm
a_{123}+\sqrt{-a_{\smash{\pm}}^{(2)}}\Bigr{)}\\\
&+\tfrac{1}{2}\log\Bigl{(}a_{0}\pm
a_{123}-\sqrt{-a_{\smash{\pm}}^{(2)}}\Bigr{)},\end{aligned}&\begin{aligned}
(a_{\pm}^{(2)}<0)\land(a_{0}\pm a_{123}>0)\end{aligned}\\\ \log(a_{0}\pm
a_{123})+2\pi c_{2\pm}\hat{\mathcal{F}},&(a_{\pm}^{(2)}=0)\land(a_{0}\pm
a_{123}>0)\\\ \log\bigl{(}-(a_{0}\pm a_{123})\bigr{)}+(\pi+2\pi
c_{2\pm})\hat{\mathcal{U}},&\begin{aligned} (a_{\pm}^{(2)}=0)&\land(a_{0}\pm
a_{123}\leq 0)\\\ &\land(\mathfrak{D}=\mathrm{True})\end{aligned}\\\
\varnothing,&\begin{aligned} &\bigl{(}(a_{\pm}^{(2)}<0)\land(a_{0}\pm
a_{123}<0)\bigr{)}\\\ &\lor\bigl{(}(a_{\pm}^{(2)}=0)\land(a_{0}\pm a_{123}\leq
0)\\\
&\hphantom{\lor\bigl{(}(a_{\pm}^{(2)}=0)\land}\land(\mathfrak{D}=\mathrm{False})\bigr{)}\end{aligned}\\\
\end{cases}\allowdisplaybreaks$ (5.3) $\displaystyle\mathsf{A}_{{1,2}_{\pm}}=$
$\displaystyle\begin{cases}\frac{1}{\sqrt{a_{\smash{\pm}}^{(2)}}}\bigl{(}\arctan(a_{0}\pm
a_{123},a_{\pm})\\\ \qquad+2\pi c_{1\pm}\bigr{)}(1\pm
I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&\begin{aligned}
&(a_{\pm}^{(2)}>0)\end{aligned}\\\ \begin{aligned}
&\frac{1}{\sqrt{-a_{\smash{\pm}}^{(2)}}}\operatorname{artanh}\bigl{(}\frac{\sqrt{-a_{\smash{\pm}}^{(2)}}}{a_{0}\pm
a_{123}}\bigr{)}\\\ &\qquad\qquad\times(1\pm
I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}\end{aligned},&\begin{aligned}
(a_{\pm}^{(2)}<0)&\land(a_{0}\pm a_{123}>0)\\\
&\land\bigl{(}-a_{\pm}^{(2)}\neq(a_{0}\pm a_{123})\bigr{)}\end{aligned}\\\
\begin{aligned} &\tfrac{1}{2}\Bigl{(}\log\bigl{(}a_{0}\pm
a_{123}+\sqrt{-a_{\smash{\pm}}^{(2)}}\bigr{)}\\\ &\ -\log\bigl{(}a_{0}\pm
a_{123}-\sqrt{-a_{\smash{\pm}}^{(2)}}\bigr{)}\Bigr{)}\\\
&\quad\times\frac{1}{\sqrt{-a_{\smash{\pm}}^{(2)}}}(1\pm
I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}\end{aligned},&\begin{aligned}
(a_{\pm}^{(2)}<0)&\land(a_{0}\pm a_{123}>0)\\\
&\land\bigl{(}-a_{\pm}^{(2)}=(a_{0}\pm a_{123})\bigr{)}\end{aligned}\\\
\frac{1}{a_{0}\pm a_{123}}(1\pm
I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)},&(a_{\pm}^{(2)}=0)\land(a_{0}\pm
a_{123}>0)\\\\[2.0pt] 0,&\begin{aligned} (a_{\pm}^{(2)}=0)&\land(a_{0}\pm
a_{123}\leq 0)\\\ &\land(\mathfrak{D}=\mathrm{True})\end{aligned}\\\\[2.0pt]
\varnothing,&\begin{aligned} &\bigl{(}(a_{\pm}^{(2)}<0)\land(a_{0}\pm
a_{123}<0)\bigr{)}\\\ &\lor\bigl{(}(a_{\pm}^{(2)}=0)\land(a_{0}\pm a_{123}\leq
0)\\\
&\hphantom{\lor\bigl{(}(a_{\pm}^{(2)}<0)}\land(\mathfrak{D}=\mathrm{False})\bigr{)}\end{aligned}\\\
\end{cases}$ (5.4)

where the upper symbol in $a_{\pm}^{(2)}$ indicates that $a_{\pm}^{(2)}$
consists of the squared coefficients $a_{i}^{2}$, $a_{ij}^{2}$ and
$a_{i}a_{ij}$.
$\hat{\mathcal{F}}=\begin{cases}\hat{\mathcal{U}},&\mathrm{if}\quad\mathfrak{D}=\mathrm{True}\\\
0,&\mathrm{if}\quad\mathfrak{D}=\mathrm{False}\end{cases}$. The logical
condition $\mathfrak{D}$ is a conjunction of outcomes of three comparisons
$\mathfrak{D}=(a_{1}=\pm a_{23})\land(a_{2}=\mp a_{13})\land(a_{3}=\mp
a_{12})\equiv\bigl{(}(a_{1}=a_{23})\land(a_{2}=-a_{13})\land(a_{3}=-a_{12})\bigr{)}\lor\bigl{(}(a_{1}=-a_{23})\land(a_{2}=a_{13})\land(a_{3}=a_{12})\bigr{)}$
that should be applied to $\mathsf{A}_{{0}_{\pm}}$ and
$\mathsf{A}_{{1,2}_{\pm}}$ terms without paying attention to $\pm$ signs in
their subscripts. Unit bivector in $\mathsf{A}_{{0}_{\pm}}$ may be
parameterized as
$\hat{\mathcal{U}}=\frac{d_{12}\mathbf{e}_{12}+d_{13}\mathbf{e}_{13}+d_{23}\mathbf{e}_{23}}{\sqrt{d_{12}^{2}-d_{13}^{2}-d_{23}^{2}}}$.
The symbol $\varnothing$ means that the solution set is empty. In all formulas
the indices and conditions (except $\mathfrak{D}$ as stated explicitly) must
be included with either all upper or with all lower signs.

The case with $f_{\pm}\neq 0$ and $a_{\pm}^{(2)}>0$ represents a generic
instance. When either $f_{\pm}=0$ or $a_{\pm}^{(2)}\leq 0$ we have the special
case. Note that in Eq. (5.2) the condition $f_{\pm}=0$ implies
$a_{\pm}^{(2)}\leq 0$. Also, observe that the condition $f_{\pm}\geq 0$
ensures automatically that a less restrictive requirement
$\operatorname{Det}(\mathsf{A})=f_{-}f_{+}\geq 0$ is fulfilled automatically.

The equations (5.3)-(5.4) are similar to Eqs (3.8)-(3.11) in
$\mathit{Cl}_{0,3}$ (see Sec. 3.2). Also, in (5.2) the expressions for scalar
coefficients $a_{\pm}=\begin{cases}\sqrt{a_{\pm}^{(2)}},&a_{\pm}^{(2)}\geq
0\\\ \sqrt{-a_{\pm}^{(2)}},&a_{\pm}^{(2)}<0\end{cases}$ are similar to Eqs
(3.6) and (3.7). The differences mainly arise at the parameter boundaries that
define the existence of MV logarithm for $\mathit{Cl}_{2,1}$.

From our earlier calculations [10] we know the conditions that ensure an
existence of MV square roots in $\mathit{Cl}_{2,1}$ algebra. Thus, we can
rewrite and use here these conditions that limit the extent of the logarithm
in Theorem 5.1. It appears that the quantities $b_{S}$ and $b_{I}$ in [10] may
be expressed in terms of multipliers $f_{+}$ and $f_{-}$ in the determinant
$D=\operatorname{Det}\mathsf{A}=f_{-}f_{+}$, where $f_{\pm}=(a_{0}\pm
a_{123})^{2}+a_{\pm}^{(2)}$, in a form
$b_{I}=\frac{1}{2}\bigl{(}f_{+}-f_{-}\bigr{)}$ and
$b_{S}=\frac{1}{2}\bigl{(}f_{+}+f_{-}\bigr{)}$. Now, note that $f_{\pm}$ enter
as arguments in log-functions of Theorem 5.1, Eq. (5.4). Therefore, the square
root existence condition $b_{S}-\sqrt{D}\geq 0$ in [10], in terms of the
logarithm problem can be rewritten as a difference of the determinant factors,
namely,
$b_{S}-\sqrt{D}\Leftrightarrow\frac{1}{2}\bigl{(}\sqrt{f_{-}}-\sqrt{f_{+}}\bigr{)}^{2}$.
Now it becomes clear that this condition is always satisfied and therefore can
be ignored, once we assume that the both factors satisfy $f_{-}\geq 0$ and
$f_{+}\geq 0$. From all this we conclude that the requirement $f_{\pm}>0$
constitutes one of the existence conditions of logarithm in Theorem 5.1. Also,
$b_{S}-\sqrt{D}=0$ is equivalent to $f_{+}=f_{-}$. This restricts the maximal
possible value of $a_{\pm}^{(2)}$. In particular,
$|a_{\pm}^{(2)}|\leq(a_{0}\pm a_{123})^{2}$. Remember, that notation
$a_{\pm}^{(2)}$ (instead of $a_{\pm}$) was introduced to keep an analogy with
$\mathit{Cl}_{0,3}$ case. It may be negative $a_{\pm}^{(2)}<0$ (see definition
(5.2)) and therefore the notation, in general, can’t be interpreted as a
square of scalar unless $a_{\pm}^{(2)}\geq 0$. When $a_{\pm}^{(2)}=0$, an
additional condition $a_{0}\pm a_{123}\geq 0$ is required for logarithm to
exist.

Since $\mathit{Cl}_{2,1}$ algebra is rarely used we will not provide explicit
formulas for pure blades (they can be found in the notebook
ElementaryFunctions.nb in [17]). Also, because generic formulas are similar to
those in $\mathit{Cl}_{0,3}$ the examples below are restricted to special
cases only.

Example 10. Logarithm in $\mathit{Cl}_{2,1}$ when $a_{\pm}^{(2)}=0$ and
$a_{0}\pm a_{123}>0$. Let the MV be
$\mathsf{A}=7+(2\mathbf{e}_{1}+\mathbf{e}_{2}+3\mathbf{e}_{3})+(2\mathbf{e}_{12}+2\mathbf{e}_{13}-2\mathbf{e}_{23})+5I=7+\mathbf{a}+\mathcal{A}+5I$.
From (5.2) we find $a_{+}^{(2)}=0,f_{+}=144$ and $a_{-}^{(2)}=0,f_{-}=144$.
Since $a_{0}\pm a_{123}=7\pm 5>0$ from (5.3) we have
$\mathsf{A}_{{0}_{-}}=\log 2$, $\mathsf{A}_{{0}_{+}}=\log 12$ and from (5.4)
$\mathsf{A}_{{1,2}_{-}}=\frac{1}{2}(1-I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}$,
$\mathsf{A}_{{1,2}_{+}}=\frac{1}{12}(1+I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}$.
Finally,
$\log\mathsf{A}=\frac{1}{24}\bigl{(}12\log(24)+24\mathbf{e}_{1}+17\mathbf{e}_{2}+31\mathbf{e}_{3}+29\mathbf{e}_{12}+19\mathbf{e}_{13}-24\mathbf{e}_{23}+12\log(6)I\bigr{)}$.
Note, because MV coefficients $a_{3}\neq\pm a_{12}$ the condition
$\mathfrak{D}$ is $\mathrm{False}$, therefore the free MV in (5.3) is absent,
$\hat{\mathcal{F}}=0$.

Example 11. Logarithm when $a_{-}^{(2)}=0$, $a_{0}-a_{123}=0$ and
$a_{+}^{(2)}>0$, $a_{0}+a_{123}<0$. In $\mathit{Cl}_{2,1}$ these properties
are satisfied by MV
$\mathsf{A}=-2+(7\mathbf{e}_{1}+4\mathbf{e}_{2}+10\mathbf{e}_{3})+(-10\mathbf{e}_{12}-4\mathbf{e}_{13}+7\mathbf{e}_{23})-2I=-2+\mathbf{a}+\mathcal{A}-2I$.
From (5.2) we find $a_{+}^{(2)}=140,f_{+}=156$ and $a_{-}^{(2)}=0,f_{-}=0$.
Then, because $a_{0}-a_{123}=-2-(-2)=0$ and $a_{0}+a_{123}=-2-2<0$, from (5.3)
we have $\mathsf{A}_{{0}_{-}}=\log(0_{+})+(\pi+2\pi
c_{2_{-}})\hat{\mathcal{U}},\mathsf{A}_{{0}_{+}}=\frac{1}{2}\log(156)$. From
(5.4)
$\mathsf{A}_{{1,2}_{-}}=0,\mathsf{A}_{{1,2}_{+}}=\frac{2}{\sqrt{35}}\bigl{(}\pi+2\pi
c_{1_{+}}-\arctan(\sqrt{35}/2)\bigr{)}(1+I)\bigl{(}\mathbf{a}+\mathcal{A}\bigr{)}$.
Then using (5.1) we obtain the final answer
$\log\mathsf{A}=\alpha_{0}+\alpha_{1}\mathbf{e}_{1}+\alpha_{2}\mathbf{e}_{2}+\alpha_{3}\mathbf{e}_{3}+\alpha_{12}\mathbf{e}_{12}+\alpha_{13}\mathbf{e}_{13}+\alpha_{23}\mathbf{e}_{23}+\alpha_{123}I$,
where $\beta=\arctan(\sqrt{35}/2)$ and,

$\begin{split}&\alpha_{0}=\tfrac{1}{2}(\log
0_{+}+\log\sqrt{156}),\quad\alpha_{123}=-\tfrac{1}{4}(2\log 0_{+}-\log
156),\\\
&\alpha_{1}=-\tfrac{1}{20}\big{(}5\sqrt{3}\,\pi-3\sqrt{35}\,\pi+2\sqrt{35}\,\beta\big{)},\quad\alpha_{2}=\big{(}\tfrac{\pi}{2\sqrt{3}}+\tfrac{2}{\sqrt{35}}\,\pi-\tfrac{2}{\sqrt{35}}\beta\big{)},\\\
&\alpha_{3}=\big{(}\sqrt{\tfrac{5}{7}}\,\pi+\tfrac{5\pi}{4\sqrt{3}}-\sqrt{\tfrac{5}{7}}\,\beta\big{)},\quad\alpha_{12}=\big{(}-\sqrt{\tfrac{5}{7}}\,\pi+\tfrac{5\pi}{4\sqrt{3}}+\sqrt{\tfrac{5}{7}}\,\beta\big{)},\\\
&\alpha_{13}=\big{(}\tfrac{\pi}{2\sqrt{3}}-\tfrac{2}{\sqrt{35}}\,\pi+\tfrac{2}{\sqrt{35}}\beta\big{)},\quad\alpha_{23}=\tfrac{1}{20}\big{(}5\sqrt{3}\,\pi+2\sqrt{35}\,\pi-2\sqrt{35}\,\beta\big{)}.\end{split}$

For simplicity the constants $c_{i}{{}_{\pm}}$ and $\hat{\mathcal{U}}$ were
equated to zero. One can check that after replacement of $\log(0_{+})$ by
$\log(x)$ and substituting the final result into exponenial formula $(23)$ in
[8] and then computing the limit $x\to 0$ we recover the initial MV. To make
the verification simple when $c_{i}{{}_{\pm}}$ and $\hat{\mathcal{U}}$ are
included, one may choose concrete values for arbitrary free constants
$c_{i}{{}_{\pm}}$ and arbitrary unit bivector $\hat{\mathcal{U}}^{2}=-1$.

##  6\. Roots and arbitrary powers of MV

If GA logarithm is known, the powers of a MV may be computed with
$\mathsf{A}^{r}=\exp\bigl{(}r\log{\mathsf{A}}\bigr{)}$, i.e., by multiplying
logarithm by power value $r$,which may be either an integer or a rational
number, and then computing the exponential. In the preprint [10] we provided
the algorithm how to obtain all possible square roots ($r=1/2$) of MV for all
$n=3$ Clifford algebras. Here we want to show that the roots presented in [10]
as numerical examples of algorithm are consistent with the above exp-log
formula, thus actually we perform a cross check of 3D GA logarithm formulas by
different methods. It should be stressed that the logarithm formula allows to
find only a single666More precisely two (plus/minus) roots. square root,
although there may exist, as shown in [10], many (up to 16 in case of
$\mathit{Cl}_{2,1}$ algebra) roots. Thus, the GA logarithm function is not
universal enough, although it may be sometimes useful if only a single
fractional root, $r=1/n$ and $n\in\mathbb{N}$, is needed.

Example 12. $\mathit{Cl}_{3,0}$. Example 1 from [10]. Theorem 4.1 is used to
calculate the root of MV $\mathsf{A}=\mathbf{e}_{1}-2\mathbf{e}_{12}$. In
$\mathit{Cl}_{3,0}$ algebra the logarithm is $\log\mathsf{A}=\frac{\log
5}{2}-\frac{1}{2}\pi\mathbf{e}_{23}+\arctan(\tfrac{1}{2})I$. Then the square
root $\sqrt{\mathsf{A}}$ is

$\displaystyle\exp\bigl{(}\tfrac{1}{2}\log\mathsf{A}\bigr{)}=$
$\displaystyle\frac{\sqrt[4]{5}}{\sqrt{2}}\bigl{(}\cos(\tfrac{1}{2}\arctan\tfrac{1}{2})\bigl{(}1-\mathbf{e}_{23}\bigr{)}+\sin(\tfrac{1}{2}\arctan\tfrac{1}{2})\bigl{(}\mathbf{e}_{1}+I\bigr{)}\bigr{)},$

which after simplification coincides with root $A_{3}$ in Example 1 in [10].

Example 13. $\mathit{Cl}_{3,0}$. Example 2 from [10]. Logarithm of MV
$\mathsf{A}=-1+\mathbf{e}_{3}-\mathbf{e}_{12}+\tfrac{1}{2}I$ in
$\mathit{Cl}_{3,0}$ is

$\displaystyle\log\mathsf{A}=$
$\displaystyle\log\bigl{(}\tfrac{\sqrt{5}}{2}\bigr{)}-\tfrac{\log
5}{2}\mathbf{e}_{3}+\tfrac{1}{2}\bigl{(}\pi-\arctan\tfrac{4}{3}\bigr{)}\mathbf{e}_{12}+\bigl{(}-\pi+\arctan\tfrac{1}{2}\bigr{)}I,$

Multiplication by $\frac{1}{2}$ and exponentiation gives the root
$A_{3}=\sqrt{\mathsf{A}}=\frac{1}{2}\bigl{(}\mathbf{e}_{3}+\mathbf{e}_{12}\bigr{)}-I$
which coincides with [10].

Example 14. $\mathit{Cl}_{3,0}$. Example 3 from [10]. Similarly, the logarithm
of MV $\mathsf{A}=-1+\mathbf{e}_{123}$ in $\mathit{Cl}_{3,0}$ is found to be
$\log\mathsf{A}=\frac{\log 2}{2}+\frac{3}{4}\pi I$. Multiplication by
$\frac{1}{2}$ and exponentiation gives the root $A_{1}$ of Example 3 [10],
$\sqrt{\mathsf{A}}=2^{1/4}\big{(}\cos\tfrac{3\pi}{8}+I\sin\tfrac{3\pi}{8}\big{)}=\sqrt{-\tfrac{1}{2}+\tfrac{1}{\sqrt{2}}}+I\sqrt{\tfrac{1}{2}+\tfrac{1}{\sqrt{2}}}$.
Likewise, an attempt to compute the logarithm of
$\mathsf{A}=\mathbf{e}_{1}+\mathbf{e}_{12}$ yields empty set, i.e., the
logarithm and as a result the square root do not exist.

Example 15. $\mathit{Cl}_{3,0}$. Quaternion. Example 4 from [10]. In
$\mathit{Cl}_{3,0}$ algebra the even MV
$\mathsf{A}=1+\mathbf{e}_{12}-\mathbf{e}_{13}+\mathbf{e}_{23}$ is equivalent
to Hamilton quaternion. The logarithm is $\log\mathsf{A}=\log
2+\frac{\pi}{3\sqrt{3}}\bigl{(}\mathbf{e}_{12}-\mathbf{e}_{13}+\mathbf{e}_{23}\bigr{)}$.
Multiplication by $\frac{1}{2}$ and exponentiation give the root $A_{3}$ in
Example 4 [10],
$\sqrt{\mathsf{A}}=\tfrac{1}{\sqrt{6}}(3+\mathbf{e}_{12}-\mathbf{e}_{13}+\mathbf{e}_{23})$.

Example 16. $\mathit{Cl}_{0,3}$. Example 6 from [10]. To compute the logarithm
of MV $\mathsf{A}=\mathbf{e}_{1}-2\mathbf{e}_{23}$ in $\mathit{Cl}_{0,3}$ the
Theorem 3.1 was applied which gives $\log\mathsf{A}=\frac{\log
3}{2}-\frac{\pi}{2}\mathbf{e}_{23}+\frac{\log 3}{2}I$. Multiplication by
$\frac{1}{2}$ and exponentiation gives the root $A_{3}$ of Example 6 in [10],
$\sqrt{\mathsf{A}}=\tfrac{1}{2}(d_{2}+d_{1}\mathbf{e}_{1}-d_{2}\mathbf{e}_{23}+I/d_{2})$,
where $d_{1}=\sqrt{2-\sqrt{3}}$ and $d_{2}=\sqrt{2+\sqrt{3}}$.

Example 17. $\mathit{Cl}_{0,3}$. Example 7 from [10]. The logarithm of MV
$\mathsf{A}=-\mathbf{e}_{3}+\mathbf{e}_{12}+4I$ in $\mathit{Cl}_{0,3}$ algebra
is computed by Theorem 3.1. The result is $\log\mathsf{A}=\frac{\log
320}{4}-\frac{1}{2}\arctan(\frac{1}{2})\mathbf{e}_{3}+\frac{1}{2}\arctan(\frac{1}{2})\mathbf{e}_{12}+\frac{\pi}{2}\bigl{(}1-I\bigr{)}\hat{\mathbf{u}}+\frac{1}{4}\log\frac{5}{4}I$.
We have assumed that the discrete free constants are equal to zero and
retained only free unit vector $\hat{\mathbf{u}}$ that satisfies
$\hat{\mathbf{u}}^{2}=-1$. Multiplication by $\frac{1}{2}$ and exponentiation
then gives
$\sqrt{\mathsf{A}}=\frac{1}{2}c_{2}+\frac{1}{2}c_{1}\bigl{(}\mathbf{e}_{12}-\mathbf{e}_{3}\bigr{)}+\frac{1}{2}c_{2}I+\bigl{(}1-I\bigr{)}\hat{\mathbf{u}}$,
where $c_{1}=\sqrt{\sqrt{5}-2}$ and $c_{2}=\sqrt{2+\sqrt{5}}$, and corresponds
to $A_{3}$ root in Example 7 in [10]. In particular, in order to obtain the
numerical value corresponding to $V_{2}=\frac{1}{2},V_{3}=0$ of [10] we have
to take
$\hat{\mathbf{u}}=-\frac{1}{2}\sqrt{5-\sqrt{5}-2c_{1}}\mathbf{e}_{1}+\frac{1}{2}(-1-c_{1})\mathbf{e}_{3}$.

Example 18. $\mathit{Cl}_{2,1}$. Example 8 from [10]. With the Theorem 5.1 one
may ascertain that the logarithm of MV
$\mathsf{A}=\mathbf{e}_{1}-2\mathbf{e}_{23}$ in $\mathit{Cl}_{2,1}$ algebra
does not exist what is in agreement with square root absence of this MV in
$\mathit{Cl}_{2,1}$. On the other hand, the logarithm of MV
$\mathsf{A}=2+\mathbf{e}_{1}+\mathbf{e}_{13}$ is $\log\mathsf{A}=\frac{\log
2}{2}-\frac{1}{\sqrt{2}}\operatorname{artanh}\bigl{(}\frac{1}{\sqrt{2}}\bigr{)}\bigl{(}\mathbf{e}_{1}+\mathbf{e}_{13}\bigr{)}$.
Multiplication by $\frac{1}{2}$ and exponentiation gives the root $A_{5}$ of
example 8 in [10],
$\sqrt{\mathsf{A}}=\frac{1}{2}\left(\sqrt{2-\sqrt{2}}\,(\mathbf{e}_{1}+\mathbf{e}_{13})+\sqrt{2\left(2+\sqrt{2}\right)}\right)$.

Of course, after multiplication of logarithm by any integer or rational number
and subsequent exponentiation we can obtain a corresponding power of the MV.
For example, in $\mathit{Cl}_{3,0}$ the logarithm of
$\mathsf{A}=\mathbf{e}_{1}$ is $\log\mathsf{A}=\frac{\pi}{2}\mathbf{e}_{1}$.
Then, it is easy to check that after multiplication by $\frac{1}{3}$ and
exponentiation we obtain the cubic root
$\sqrt[3]{\mathbf{e}_{1}}=\frac{1}{2}\bigl{(}\sqrt{3}+\mathbf{e}_{1}\bigr{)}$.

##  7\. Relations of the logarithm to GA inverse trigonometric and hyperbolic
functions

Just like trigonometric and hyperbolic functions can be expressed by
exponentials (Euler and de Moivre formulas), the inverse hyperbolic functions
may be defined in terms of logarithms. Therefore, in GA we can use the
following definitions to compute inverse hyperbolic and trigonometric
functions of MV argument $\mathsf{A}$.
For hyperbolic inverse functions:

$\displaystyle\operatorname{artanh}\mathsf{A}=$
$\displaystyle\tfrac{1}{2}\bigl{(}\log(1+\mathsf{A})-\log(1-\mathsf{A})\bigr{)},$
(7.1) $\displaystyle\operatorname{arcoth}\mathsf{A}=$
$\displaystyle\begin{cases}\frac{1}{2}\bigl{(}\log(1+\mathsf{A}^{-1})-\log(1-\mathsf{A}^{-1})\bigr{)},&\mathsf{A}\neq
0,\\\ \frac{\pi}{2}I,&\mathsf{A}=0,\end{cases}\allowdisplaybreaks$ (7.2)
$\displaystyle\operatorname{arcosh}\mathsf{A}=$
$\displaystyle\log\bigl{(}\mathsf{A}+\sqrt{\mathsf{A}-1}\,\sqrt{\mathsf{A}+1}\bigr{)},$
(7.3) $\displaystyle\operatorname{arsinh}\mathsf{A}=$
$\displaystyle\log\bigl{(}\mathsf{A}+\sqrt{\mathsf{A}^{2}+1}\bigr{)}.$ (7.4)

For inverse trigonometric functions:

$\displaystyle\arcsin\mathsf{A}=$
$\displaystyle-I\log\bigl{(}\mathsf{A}I+\sqrt{1-\mathsf{A}^{2}}\bigr{)}\allowdisplaybreaks,$
(7.5) $\displaystyle\arccos\mathsf{A}=$
$\displaystyle\frac{\pi}{2}+I\log\bigl{(}\mathsf{A}I+\sqrt{1-\mathsf{A}^{2}}\bigr{)},$
(7.6) $\displaystyle\arctan\mathsf{A}=$
$\displaystyle\frac{I}{2}\bigl{(}\log(1-I\mathsf{A})-\log(1+I\mathsf{A})\bigr{)}\allowdisplaybreaks,$
(7.7) $\displaystyle\operatorname{arccot}\mathsf{A}=$
$\displaystyle\begin{cases}\frac{1}{2}I\bigl{(}\log(1-I\mathsf{A}^{-1})-\log(1+I\mathsf{A}^{-1})\bigr{)},&\mathsf{A}\neq
0,\\\ \frac{\pi}{2},&\mathsf{A}=0.\end{cases}$ (7.8)

These formulas are similar to those in the theory of real and complex
functions except that instead of the imaginary unit the pseudoscalar appears
in trigonometric functions. However, earlier we have found [10] that in GA the
functions with the square root, in general, are multi-valued. Thus at a first
sight it may appear that the listed above equations with square root are not
valid in all circumstances. Nonetheless, our preliminary numerical experiments
show that they, in fact, are satisfied for all possible individual plus/minus
pairs of square roots777This property does not allow us to write the equality
sign between GA general expression $\log\sqrt{\mathsf{B}}$ and
$\frac{1}{2}\log\mathsf{B}$. (see Example 8 in [8] and Example 19 below in
this section).

With the above formulas for hyperbolic and trigonometric functions one can
construct the following identities for generic MVs:888Trigonometric functions
are defined only for algebras where the pseudoscalar 1) belongs to a center of
an algebra, i.e. commutes commutative with remaining elements and 2) satisfy
$I^{2}=-1$. In the considered 3D algebras only for $\mathit{Cl}_{3,0}$ and
$\mathit{Cl}_{1,2}$.

$\displaystyle\sinh\mathsf{A}=$
$\displaystyle\tfrac{1}{2}\bigl{(}\exp(\mathsf{A})-\exp(-\mathsf{A})\bigr{)},$
(7.9) $\displaystyle\cosh\mathsf{A}=$
$\displaystyle\tfrac{1}{2}\bigl{(}\exp(\mathsf{A})+\exp(-\mathsf{A})\bigr{)}\allowdisplaybreaks,$
(7.10) $\displaystyle\tanh\mathsf{A}=$
$\displaystyle\sinh\mathsf{A}(\cosh\mathsf{A})^{-1}=\bigl{(}\exp(\mathsf{A})-\exp(-\mathsf{A})\bigr{)}\bigl{(}\exp(\mathsf{A})+\exp(-\mathsf{A})\bigr{)}^{-1},$
(7.11) $\displaystyle\coth\mathsf{A}=$
$\displaystyle\cosh\mathsf{A}(\sinh\mathsf{A})^{-1}=\bigl{(}\exp(\mathsf{A})+\exp(-\mathsf{A})\bigr{)}\bigl{(}\exp(\mathsf{A})-\exp(-\mathsf{A})\bigr{)}^{-1}\allowdisplaybreaks.$
(7.12) $\displaystyle\sin\mathsf{A}=$
$\displaystyle\tfrac{1}{2}I^{-1}\bigl{(}\exp(I\mathsf{A})-\exp(-I\mathsf{A})\bigr{)},$
(7.13) $\displaystyle\cos\mathsf{A}=$
$\displaystyle\tfrac{1}{2}\bigl{(}\exp(I\mathsf{A})+\exp(-I\mathsf{A})\bigr{)}\allowdisplaybreaks,$
(7.14) $\displaystyle\tan\mathsf{A}=$
$\displaystyle\sin\mathsf{A}(\cos\mathsf{A})^{-1}=-I\bigl{(}\exp(I\mathsf{A})-\exp(-I\mathsf{A})\bigr{)}\bigl{(}\exp(I\mathsf{A})+\exp(-I\mathsf{A})\bigr{)}^{-1},$
(7.15) $\displaystyle\cot\mathsf{A}=$
$\displaystyle\cos\mathsf{A}(\sin\mathsf{A})^{-1}=I\bigl{(}\exp(I\mathsf{A})+\exp(-I\mathsf{A})\bigr{)}\bigl{(}\exp(I\mathsf{A})-\exp(-I\mathsf{A})\bigr{)}^{-1}.$
(7.16)

We have not investigated how the presented formulas work in case when the MV
square root or logarithm allows answer that depends on non-discrete free
parameters and when the inverse MVs can’t be be computed. Also, we have not
considered MV logarithms that allow infinite coefficients at some of basis
MVs.

Example 19. Inverse MV hyperbolic functions. To save space we will restrict
ourselves to numerical examples only for $\mathit{Cl}_{3,0}$ generic MV
$\mathsf{A}=-1-5\mathbf{e}_{1}+7\mathbf{e}_{2}-9\mathbf{e}_{3}+7\mathbf{e}_{12}-5\mathbf{e}_{13}+9\mathbf{e}_{23}+9I$.
Then we find the following inverse hyperbolic functions,

$\displaystyle\operatorname{artanh}\mathsf{A}=$ $\displaystyle\kern
10.00002pt\begin{aligned}
&0.0544776&\kern-10.00002pt-&0.0683983\mathbf{e}_{1}&\kern-10.00002pt-&0.0034179\mathbf{e}_{2}&\kern-10.00002pt+&0.0712752\mathbf{e}_{3}\\\
\kern-10.00002pt-&0.0259578\mathbf{e}_{12}&\kern-10.00002pt-&0.0571283\mathbf{e}_{13}&\kern-10.00002pt+&0.0036554\mathbf{e}_{23}&\kern-10.00002pt+&1.5447402I,\end{aligned}$
$\displaystyle\operatorname{arcoth}\mathsf{A}=$ $\displaystyle\kern
10.00002pt\begin{aligned}
&0.0544776&\kern-10.00002pt-&0.0683983\mathbf{e}_{1}&\kern-10.00002pt-&0.0034179\mathbf{e}_{2}&\kern-10.00002pt-&0.0712752\mathbf{e}_{3}\\\
\kern-10.00002pt-&0.0259578\mathbf{e}_{12}&\kern-10.00002pt-&0.0571283\mathbf{e}_{13}&\kern-10.00002pt+&0.0036555\mathbf{e}_{23}&\kern-10.00002pt-&0.0260523I,\end{aligned}$
$\displaystyle\operatorname{arcosh}\mathsf{A}=$ $\displaystyle\kern
10.00002pt\begin{aligned}
&3.1995844&\kern-10.00002pt+&0.6349751\mathbf{e}_{1}&\kern-10.00002pt+&0.6477695\mathbf{e}_{2}&\kern-10.00002pt+&0.3396621\mathbf{e}_{3}\\\
\kern-10.00002pt+&0.9970274\mathbf{e}_{12}&\kern-10.00002pt+&0.4603461\mathbf{e}_{13}&\kern-10.00002pt+&0.7081115\mathbf{e}_{23}&\kern-10.00002pt+&1.0647020I.\end{aligned}$

For identities that contain square roots $\sqrt{\mathsf{A}\pm 1}$, for example
$\operatorname{arcosh}\mathsf{A}$ or $\operatorname{arsinh}\mathsf{A}$, all
four roots are valid. Below they have been calculated by algorithm described
in [10],

$\displaystyle\textit{Root 1 and 2}:$ $\displaystyle\sqrt{\mathsf{A}-1}=\pm($
$\displaystyle\kern 10.00002pt\begin{aligned}
\kern-10.00002pt-&2.3936546&\kern-10.00002pt-&0.3144420\mathbf{e}_{1}&\kern-10.00002pt-&1.3708134\mathbf{e}_{2}&\kern-10.00002pt+&0.3806804\mathbf{e}_{3}\\\
\kern-10.00002pt-&1.7824116\mathbf{e}_{12}&\kern-10.00002pt-&0.1086429\mathbf{e}_{13}&\kern-10.00002pt-&1.6154750\mathbf{e}_{23}&\kern-10.00002pt-&2.0134421I),\end{aligned}$
$\displaystyle\indent\textit{Root 3 and 4}:$
$\displaystyle\sqrt{\mathsf{A}-1}=\pm($ $\displaystyle\kern
10.00002pt\begin{aligned}
\kern-10.00002pt-&0.1660207&\kern-10.00002pt+&2.4324037\mathbf{e}_{1}&\kern-10.00002pt+&1.1337774\mathbf{e}_{2}&\kern-10.00002pt+&2.0055931\mathbf{e}_{3}\\\
\kern-10.00002pt+&2.1654007\mathbf{e}_{12}&\kern-10.00002pt+&1.9165921\mathbf{e}_{13}&\kern-10.00002pt+&1.0892769\mathbf{e}_{23}&\kern-10.00002pt+&1.9243691I).\end{aligned}$

And similarly for

$\displaystyle\sqrt{\mathsf{A}+1}=\pm\\{$ $\displaystyle\kern
10.00002pt\begin{aligned}
\kern-10.00002pt-&2.6330243&\kern-10.00002pt-&0.1908183\mathbf{e}_{1}&\kern-10.00002pt-&1.3218252\mathbf{e}_{2}&\kern-10.00002pt+&0.4871255\mathbf{e}_{3}\\\
\kern-10.00002pt-&1.6829550\mathbf{e}_{12}&\kern-10.00002pt-&0.0102534\mathbf{e}_{13}&\kern-10.00002pt-&1.5705147\mathbf{e}_{23}&\kern-10.00002pt-&1.9117486I\\},\\\\[5.0pt]
\end{aligned}$ $\displaystyle\sqrt{\mathsf{A}+1}=\pm\\{$ $\displaystyle\kern
10.00002pt\begin{aligned}
\kern-10.00002pt-&0.2910283&\kern-10.00002pt+&2.6047118\mathbf{e}_{1}&\kern-10.00002pt+&1.0343473\mathbf{e}_{2}&\kern-10.00002pt+&2.2416242\mathbf{e}_{3}\\\
\kern-10.00002pt+&2.0981982\mathbf{e}_{12}&\kern-10.00002pt+&2.0727864\mathbf{e}_{13}&\kern-10.00002pt+&0.9499284\mathbf{e}_{23}&\kern-10.00002pt+&1.8337753I\\}.\end{aligned}$

It is important to stress that, in general, the individual formulas
($\arccos\mathsf{A}$, $\arcsin\mathsf{A}$ and their hyperbolic analogues) that
contain the sets of roots yield different function values for four different
roots in the above listed sets.

$\displaystyle\operatorname{arsinh}\mathsf{A}=\pm($ $\displaystyle\kern
10.00002pt\begin{aligned} \kern-10.00002pt\
&3.2035891&\kern-10.00002pt+&0.6313828\mathbf{e}_{1}&\kern-10.00002pt+&0.6490577\mathbf{e}_{2}&\kern-10.00002pt+&0.3351515\mathbf{e}_{3}\\\
\kern-10.00002pt+&0.9974654\mathbf{e}_{12}&\kern-10.00002pt+&0.4571790\mathbf{e}_{13}&\kern-10.00002pt+&0.7100715\mathbf{e}_{23}&\kern-10.00002pt+&1.0647010I).\\\
\end{aligned}$ $\displaystyle\operatorname{arsinh}\mathsf{A}=\pm($
$\displaystyle\kern 10.00002pt\begin{aligned} \kern-10.00002pt\
&0.4835482&\kern-10.00002pt+&2.5061943\mathbf{e}_{1}&\kern-10.00002pt-&0.7303556\mathbf{e}_{2}&\kern-10.00002pt+&3.0588480\mathbf{e}_{3}\\\
\kern-10.00002pt-&0.0989201\mathbf{e}_{12}&\kern-10.00002pt+&2.1904765\mathbf{e}_{13}&\kern-10.00002pt-&1.1645414\mathbf{e}_{23}&\kern-10.00002pt+&2.5756463I).\end{aligned}$

Example 20. Inverse trigonometric functions of MV. Numerical answers for
$\mathit{Cl}_{3,0}$ generic MV
$\mathsf{A}=-1-5\mathbf{e}_{1}+7\mathbf{e}_{2}-9\mathbf{e}_{3}+7\mathbf{e}_{12}-5\mathbf{e}_{13}+9\mathbf{e}_{23}+9I$
in a form of list for roots 1-4,

$\displaystyle\arcsin\mathsf{A}=\\{$ $\displaystyle\kern
10.00002pt\begin{aligned} \kern-10.00002pt\
&2.5745928&\kern-10.00002pt+&0.1233316\mathbf{e}_{1}&\kern-10.00002pt-&2.3715122\mathbf{e}_{2}&\kern-10.00002pt+&1.3713947\mathbf{e}_{3}\\\
\kern-10.00002pt-&2.8712504\mathbf{e}_{12}&\kern-10.00002pt+&0.3732007\mathbf{e}_{13}&\kern-10.00002pt-&2.8706092\mathbf{e}_{23}&\kern-10.00002pt-&0.4882339I,\\\
\kern-10.00002pt\
&2.6354984&\kern-10.00002pt+&0.7081116\mathbf{e}_{1}&\kern-10.00002pt-&0.4603462\mathbf{e}_{2}&\kern-10.00002pt+&0.9970274\mathbf{e}_{3}\\\
\kern-10.00002pt-&0.3396621\mathbf{e}_{12}&\kern-10.00002pt+&0.6477695\mathbf{e}_{13}&\kern-10.00002pt-&0.6349751\mathbf{e}_{23}&\kern-10.00002pt-&3.1995845I,\\\
\kern-10.00002pt+&0.5669998&\kern-10.00002pt-&0.1233316\mathbf{e}_{1}&\kern-10.00002pt+&2.3715122\mathbf{e}_{2}&\kern-10.00002pt-&1.3713947\mathbf{e}_{3}\\\
\kern-10.00002pt+&2.8712504\mathbf{e}_{12}&\kern-10.00002pt-&0.3732007\mathbf{e}_{13}&\kern-10.00002pt+&2.8706092\mathbf{e}_{23}&\kern-10.00002pt+&0.4882339I,\\\
\kern-10.00002pt\
&0.5060943&\kern-10.00002pt-&0.7081116\mathbf{e}_{1}&\kern-10.00002pt+&0.4603462\mathbf{e}_{2}&\kern-10.00002pt-&0.9970274\mathbf{e}_{3}\\\
\kern-10.00002pt+&0.3396621\mathbf{e}_{12}&\kern-10.00002pt-&0.6477695\mathbf{e}_{13}&\kern-10.00002pt+&0.6349751\mathbf{e}_{23}&\kern-10.00002pt+&3.1995845I\\}.\end{aligned}$

Since formulas for arc sine and cosine also include square roots we obtain
four different values for these functions too,

$\displaystyle\arccos\mathsf{A}=\pm\\{$ $\displaystyle\kern
10.00002pt\begin{aligned}
\kern-10.00002pt-&1.0037965&\kern-10.00002pt-&0.1233316\mathbf{e}_{1}&\kern-10.00002pt+&2.3715122\mathbf{e}_{2}&\kern-10.00002pt-&1.3713947\mathbf{e}_{3}\\\
\kern-10.00002pt+&2.8712504\mathbf{e}_{12}&\kern-10.00002pt-&0.3732007\mathbf{e}_{13}&\kern-10.00002pt+&2.8706092\mathbf{e}_{23}&\kern-10.00002pt+&0.4882339I,\\\
\kern-10.00002pt-&1.0647021&\kern-10.00002pt-&0.7081116\mathbf{e}_{1}&\kern-10.00002pt+&0.4603462\mathbf{e}_{2}&\kern-10.00002pt-&0.9970274\mathbf{e}_{3}\\\
\kern-10.00002pt+&0.3396621\mathbf{e}_{12}&\kern-10.00002pt-&0.6477695\mathbf{e}_{13}&\kern-10.00002pt+&0.6349751\mathbf{e}_{23}&\kern-10.00002pt+&3.1995845I\\}.\end{aligned}$

On the other hand the trigonometric tangent and cotangent have a single value
since the square root here is absent, Eqs (7.7) and (7.8),

$\displaystyle\arctan\mathsf{A}=$ $\displaystyle\kern
10.00002pt\begin{aligned} \kern-10.00002pt\
&1.5171201&\kern-10.00002pt+&0.0678435\mathbf{e}_{1}&\kern-10.00002pt+&0.0036019\mathbf{e}_{2}&\kern-10.00002pt+&0.0705863\mathbf{e}_{3}\\\
\kern-10.00002pt+&0.0260071\mathbf{e}_{12}&\kern-10.00002pt+&0.0566409\mathbf{e}_{13}&\kern-10.00002pt-&0.0033708\mathbf{e}_{23}&\kern-10.00002pt+&0.0259164I,\end{aligned}$
$\displaystyle\operatorname{arccot}\mathsf{A}=$ $\displaystyle\kern
10.00002pt\begin{aligned} \kern-10.00002pt\
&0.0536762&\kern-10.00002pt-&0.0678435\mathbf{e}_{1}&\kern-10.00002pt-&0.0036019\mathbf{e}_{2}&\kern-10.00002pt-&0.0705863\mathbf{e}_{3}\\\
\kern-10.00002pt-&0.0260071\mathbf{e}_{12}&\kern-10.00002pt-&0.0566409\mathbf{e}_{13}&\kern-10.00002pt+&0.0033708\mathbf{e}_{23}&\kern-10.00002pt-&0.0259164I.\end{aligned}$

##  8\. Discussion and conclusions

The logarithm together with the exponential [8, 7] and square root [10] are
the most important functions in Clifford geometric algebra (GA). Starting from
the respective exponential functions we presented here, as far as we know, for
the first time the basis-free formulas for logarithms in all 3D GAs. The
formulas for both the generic and special cases may be directly applied in GA
programming. They were cross-checked using the basis-free GA exponential
functions found in [8]. The derived formulas were implemented in Mathematica
and tested with thousands of randomly generated multivectors [17]. In all
cases the exponentiation of the logarithm was found to simplify to the initial
MV.

Using numerical experiments [17] we observed that, in accord with the
suggestion in [1], the principal value of the logarithm can be defined as a GA
logarithm having the smallest determinant norm. In almost all cases the
principal MV logarithm is attained by setting arbitrary integer parameters
$c_{i}$ in generic logarithms (Theorems 3.1, 4.1, 5.1) to zero. Exceptions
from this rule, however, may occur in the case of simple specific MVs, for
which commuting MVs may exist ( Secs. 3.3 and 4.3), and therefore not
restricted by free MVs (Eqs (3.13) and (4.13)). Apart from discrete parameters
$c_{i}$, we have also found that continuous parameters represented by free
unit vectors $\hat{\mathbf{u}}$ or bivectors $\hat{\mathcal{U}}$ may be
included in special cases as well. The parameters vanish after exponentiation
of the logarithm and do not contribute to the MV norm. Recently we have found
that such free parameters may be also introduced into lower dimensional,
quaternionic-type Clifford algebras [4]. However, more investigations are
needed in this direction.

Also, the relation between the GA logarithm and square root of MV was
investigated. The known formula
$\sqrt{\mathsf{A}}=\exp\bigl{(}\frac{1}{2}\log(\mathsf{A})\bigr{)}$ served as
an additional check correctness of GA logarithms. Unfortunately, the formula
allows to compute only a single square root from many possible roots that may
exist in GA [10]. Nevertheless, such a comparison was found to be very useful
for testing purposes. Indeed, a test of square root of a MV is an algebraic
problem since it reduces to a solution of system of algebraic equations. On
the other hand, inversion of exponential used in finding the GA logarithm in
the present paper requires solving a system of transcendental equations
(Appendix A), a problem which is much more difficult (but at the same time
more general) task. The mentioned exp-log relation also allows to check the
condition whether the MV logarithm exists at all. Indeed, since we know how to
calculate GA exponential [10] of arbitrary MV multiplied by factor
$\frac{1}{2}$, from this follows that it is $\log(\mathsf{A})$ function which
determines the existence condition for $\sqrt{\mathsf{A}}$ to exist. As a
test, we have checked using our algorithm [10] that for each MV indeed there
exists a single square root that is in agreement with the identity
$\exp\bigl{(}\frac{1}{2}\log(\mathsf{A})\bigl{)}=\sqrt{\mathsf{A}}$.

In conclusion, in the present paper the basis-free expressions have been found
for GA logarithms in all 3D real algebras. The logarithm was found to exist
for all MVs in case of real $\mathit{Cl}_{0,3}$ algebra. In Clifford algebra
$\mathit{Cl}_{3,0}$ (and $\mathit{Cl}_{1,2}$) the logarithm exists for almost
all MVs, except very small MV class which satisfies the condition
$(a_{+}^{2}+a_{-}^{2}=0)\land(a_{0}^{2}+a_{123}^{2}=0)$. For example, the
logarithm of MV $\mathbf{e}_{1}\pm\mathbf{e}_{12}$ cannot be computed in
Euclidean $\mathit{Cl}_{3,0}$ algebra. On the other hand in
$\mathit{Cl}_{2,1}$ algebra the GA logarithm is absent in large sectors of a
real coefficient space.

## References

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##  Appendix A Outline of derivation of generic logarithm formula in a
coordinate form in $\mathit{Cl}_{0,3}$

The MV exponential
$\exp\mathsf{A}=\mathsf{B}=b_{0}+\mathbf{b}+\mathcal{B}+b_{123}I$, where
$\mathbf{b}=b_{1}\mathbf{e}_{1}+b_{2}\mathbf{e}_{2}+b_{3}\mathbf{e}_{3}$ and
$\mathcal{B}=b_{12}\mathbf{e}_{12}+b_{13}\mathbf{e}_{13}+b_{23}\mathbf{e}_{23}$,
in the coordinate form was constructed in [7]. For completeness, below we
reproduce the expressions for scalar coefficients of $\mathsf{B}$,

$\displaystyle\begin{aligned}
\phantom{{}_{99}}b_{0}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\bigl{(}\mathrm{e}^{a_{123}}\cos
a_{+}+\mathrm{e}^{-a_{123}}\cos a_{-}\bigr{)},\\\
b_{123}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\bigl{(}\mathrm{e}^{a_{123}}\cos
a_{+}-\mathrm{e}^{-a_{123}}\cos
a_{-}\bigr{)},\end{aligned}\allowdisplaybreaks$ (A.1)
$\displaystyle\begin{aligned}
\phantom{{}_{99}}b_{1}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}\mathrm{e}^{a_{123}}(a_{1}-a_{23})\frac{\sin
a_{+}}{a_{+}}+\mathrm{e}^{-a_{123}}(a_{1}+a_{23})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\\\
\phantom{{}_{99}}b_{2}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}\mathrm{e}^{a_{123}}(a_{2}+a_{13})\frac{\sin
a_{+}}{a_{+}}+\mathrm{e}^{-a_{123}}(a_{2}-a_{13})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\\\
\phantom{{}_{99}}b_{3}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}\mathrm{e}^{a_{123}}(a_{3}-a_{12})\frac{\sin
a_{+}}{a_{+}}+\mathrm{e}^{-a_{123}}(a_{3}+a_{12})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\end{aligned}\allowdisplaybreaks$ (A.2)
$\displaystyle\begin{aligned}
\phantom{{}_{9}}b_{12}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}-\mathrm{e}^{a_{123}}(a_{3}-a_{12})\frac{\sin
a_{+}}{a_{+}}+\mathrm{e}^{-a_{123}}(a_{3}+a_{12})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\\\
\phantom{{}_{9}}b_{13}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}\mathrm{e}^{a_{123}}(a_{2}+a_{13})\frac{\sin
a_{+}}{a_{+}}-\mathrm{e}^{-a_{123}}(a_{2}-a_{13})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\\\
\phantom{{}_{9}}b_{23}&=\tfrac{1}{2}\mathrm{e}^{a_{0}}\Bigl{(}-\mathrm{e}^{a_{123}}(a_{1}-a_{23})\frac{\sin
a_{+}}{a_{+}}+\mathrm{e}^{-a_{123}}(a_{1}+a_{23})\frac{\sin
a_{-}}{a_{-}}\Bigr{)},\end{aligned}$ (A.3)

where $a_{-}$ and $a_{+}$ are defined in Eqs (3.6) and (3.7), respectively.
Explicit generic formula for a MV logarithm $\text{log}\mathsf{B}=\mathsf{A}$
can be derived by inverting the equations (A.1)-(A.3). To this end, using
(A.1)-(A.3) we will construct a system of eight nonlinear trigonometric
equations for coefficients at corresponding basis elements. Since two of the
coefficients, $a_{0}$ and $a_{123}$, stand alone, i.e., they are not
associated with remaining ones in (3.6) and (3.7), as a first step in this
procedure we partially solve a pair of equations at scalar and pseudoscalar,
$b_{0}$ and $b_{123}$, to get two new functions

$\displaystyle a_{0}=$ $\displaystyle
g^{\prime}_{0}(b_{0},b_{123},a_{-},a_{+}),$ (A.4a) $\displaystyle a_{123}=$
$\displaystyle g^{\prime}_{123}(b_{0},b_{123},a_{-},a_{+}),$ (A.4b)

where the pair $(a_{-},a_{+})$ includes only vector and bivector components as
(3.6) and Eqs (3.7) show. After substituting new $a_{0}$ and $a_{123}$ into
remaining equations (A.2) and (A.3) we obtain a system of nonlinear
trigonometric equations for coefficients at vector and bivector only which,
unfortunately, cannot be solved by computer algebra system in this form.
Further progress can be achieved by observing that the following relation
between the coefficients holds in the generic case

$\displaystyle a_{-}=$ $\displaystyle\arctan(b_{0}-b_{123},b_{-}),$ (A.5a)
$\displaystyle a_{+}=$ $\displaystyle\arctan(b_{0}+b_{123},b_{+}),$ (A.5b)

where the pair $(b_{-},b_{+})$ is given by Eqs (3.7) and (3.6) after
replacement of $\mathbf{a}$ and $\mathcal{A}$, by vector $\mathbf{b}$ and
bivector $\mathcal{B}$, respectively. Then, replacing all occurrences of
$(a_{-},a_{+})$ by (A.5a) and (A.5b) in (A.2) and (A.3) (with already replaced
$a_{0}$ and $a_{123}$ as was described above) we obtain a linear system for
coefficients $a_{i},a_{ij}$, that can be solved in a straightforward way.
Finally, after substituting $a_{-}$ and $a_{+}$ as given by (A.5a) and (A.5b)
into (A.4a) and(A.4b) and performing simplifications we find the scalar
coefficients of
$\log\mathsf{B}=\mathsf{A}=a_{0}+\mathbf{a}+\mathcal{A}+a_{123}I$ expressed in
terms of two-argument arc tangent functions (see Subsec. 2.1),

$\displaystyle a_{0}=$
$\displaystyle\tfrac{1}{4}\left(\log\left((b_{0}-b_{123})^{2}+b_{-}^{2}\right)+\log\left((b_{0}+b_{123})^{2}+b_{+}^{2}\right)\right),$
$\displaystyle a_{123}=$
$\displaystyle\tfrac{1}{4}\left(\log\left((b_{0}+b_{123})^{2}+b_{+}^{2}\right)-\log\left((b_{0}-b_{123})^{2}+b_{-}^{2}\right)\right),$
$\displaystyle a_{1}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{1}+b_{23})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{1}-b_{23})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right),$
$\displaystyle a_{2}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{2}-b_{13})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{2}+b_{13})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right),$
$\displaystyle a_{3}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{3}+b_{12})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{3}-b_{12})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right),$
$\displaystyle a_{12}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{3}+b_{12})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{12}-b_{3})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right),$
$\displaystyle a_{13}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{13}-b_{2})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{2}+b_{13})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right),$
$\displaystyle a_{23}=$
$\displaystyle\tfrac{1}{2}\left(\frac{(b_{1}+b_{23})\arctan\left(b_{0}-b_{123},b_{-}\right)}{b_{-}}+\frac{(b_{23}-b_{1})\arctan\left(b_{0}+b_{123},b_{+}\right)}{b_{+}}\right).$
(A.8)

Here $b_{+}$ and $b_{-}$ are given by (3.7) and (3.6) after replacement of
$\mathbf{a}$ and $\mathcal{A}$, respectively, by vector $\mathbf{b}$ and
bivector $\mathcal{B}$. The equations (A)-(A) provide a generic solution of
the inverse problem for MV logarithm in a coordinate form. It must be
remembered that in the two-argument arc tangent functions the argument order
and properties follow Mathematica convention (see Subsec. 2.1). To have a
general GA logarithm formula of Theorem 4.1 in a basis-free form, it is enough
to multiply the coefficients in (A)-(A) by respective basis elements, to add
and regroup the resulting GA expression into coordinate-free form.