Combining strictly positive numbers to complex powers?

Question

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

$$a^{z_1}b^{z_1}=(ab)^{z_1}~~~,~~~a^{z_1}a^{z_2}=a^{z_1+z_2}$$

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

Thanks for any suggestion!

Answer 1

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.