Can we say that everywhere where it makes sense $\log_0 x=0^x$? Are they equal, the function is self-inverse? If so, what is deep intuition behind it? [closed]

Question

It makes little reason to speak about $0^x$ and $\log_0 x$ on the set of real numbers, but in matrices, it seems, the expressions coincide, for instance,

$0^ \left( \begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \\ \end{array} \right)=\left( \begin{array}{cc} \frac{1}{2} & -\frac{1}{2} \\ -\frac{1}{2} & \frac{1}{2} \\ \end{array} \right)=\log_0 \left( \begin{array}{cc} \frac{1}{2} & \frac{1}{2} \\ \frac{1}{2} & \frac{1}{2} \\ \end{array} \right),$

where $0$ is meant to be zero matrix. Since all hypercomplex numbers can be represented in matrix form, logarithm with base zero makes sense there as well, and its equivalence to the powers of zero still holds. For instance, in split-complex numbers we have

$0^{1/2+j/2}=1/2-j/2= \log_0 (1/2+j/2),$

$0^{1/2-j/2}=1/2+j/2= \log_0 (1/2-j/2).$

As such, I wonder, whether it would be sensible to claim $\log_0 x=0^x$ in all other contexts? Is there some proof or expansion that would show this is the case?

I asked a similar question on math.stackexchange but was heavily downvoted and the question was closed because, as some users claimed "logarithm with zero base has no sense".

Answer 1

Experimenting with Maple, it seems Maple's definition of $0^A$, where $A$ is a square matrix, will be:

$\bullet\;$If $A$ is diagonalizable, Say $A = Q^{-1} D Q$ with $D = \operatorname{diag}(a_1,\dots,a_n)$, then $$ 0^D = \operatorname{diag}\big(0^{a_1},\dots,0^{a_n}\big)\\ 0^A = Q^{-1} 0^D Q $$ Here $0^x = 0$ if $\operatorname{Re}(x) > 0$ and $0^0 = 1$. Some others like $0^{-1}$ and $0^i$ are undefined.

$\bullet\;$ If $A$ is not diagonalizable, then $0^A$ is undefined. For example $0^A$ is undefined if $$ A = \begin{bmatrix}0 & 1\\ 0 & 0\end{bmatrix} , $$ even though it is a zero divisor. [Hidden reason: $0^x$ is not differentiable at $x=0$.]

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I did not get Maple to do anything with $\log_0 A$.