Given complex numbers $z_1,z_2\in\mathbb{C}$ and strictly positive numbers $a,b>0$, are the following statements true in general?


If so, how to prove it? If not, which restriction would have to be applied to make it true?

Thanks for any suggestion!


Using the standard $$ e^z=\sum_{k=0}^\infty\frac{z^n}{n!} $$ and the real valued $\log(a)$ and $\log(b)$, we can define in the standard way $$ a^z=e^{z\log(a)} $$ and $$ b^z=e^{z\log(b)} $$ With theses definitions, your equations are valid.


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