Given any $z \in \mathbb C$, I know I can define $log z$ as a function once the branch is fixed. But "the base" is actually $e$, the base of natural log $ln$ in real number system. I just wonder if I can change "the base" to be any complex number. For example, define $z^{log_z w}=w$ ,$\forall z,w \in \mathbb C$. Is it possible? Are the identity concerning about log in real sense the same as those in complex sense?
Can we define something like $\log_z w$ for z,w to be complex numbers
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complex-analysis
1 Answers
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Well, if we use that:
$$\log_\text{a}\left(\text{b}\right)=\frac{\ln\text{b}}{\ln\text{a}}\tag1$$
So, when $\space\exists\space\text{a}\space\wedge\space\text{b}\in\mathbb{C}$:
$$\log_\text{a}\left(\text{b}\right)=\frac{\ln\text{b}}{\ln\text{a}}=\frac{\ln\left|\text{b}\right|+\arg\left(\text{b}\right)i}{\ln\left|\text{a}\right|+\arg\left(\text{a}\right)i}\tag2$$
Using the definiton of the complex log.