Is it possible to have Logarithm with base 1 or 0?

Question

I am wondering is there any definition that allows logarithm to have base 0 or 1 in real or complex fields (considering Euclidean space)??

Out-coming question is if you can define a logarithm with negative base??

Answer 1

In the real numbers, the logarithm is defined as the inverse of the exponential function. The functions $f: x\mapsto 1^x$ and $g:x\mapsto 0^x$ are not invertible, so the quick answer is no.

The longer answer is still no.

The problem is that the fundamental property of any logarithm function is that it satisfies the equation

$$a^{\log_a x} = x$$ for (at least) all values of $x\in(0,\infty)$ (this can further be expanded if we are looking at the complex numbers, but at the very least, the equality should hold on $(0,\infty)$). However, if $a=1$, then no matter what $\log_1 x$ is, for all values of $x$, the expression $$1^{\log_1 x}$$ will be equal to $1$, which, if $x\neq 1$, is not equal to $x$.

I would argue that any function which does not satisfy the equation $$a^{\log_a x} = x$$ is not really a logarithm function in any real sense, thus, the logarithm function for base $1$ does not exist.