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I am looking for an example of showing the following case in the complex plane. If $z^w=\exp(w\log(z))$ where $z$ is not a non-positive real number and $\log$ is the principal branch, then even if we have $\exp(\log(d))=d$ for arbitrary $d$ from the domain of $\log$ we could still have that $\log[\exp(w\log(z))]$ is not equal to $w\log(z)$.

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    I'm sorry, what kind of answer are you looking for here? It's not clear what complex function you want an example of. – 2017-01-21
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    Thank you for your reply and correction. I have edited my question accordingly. – 2017-01-21
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    As far as I know the logarithm's principal branch is $\;\Bbb C\setminus\Bbb R_{\le0}\;$ , so then what do you want to do with *non-positive reals* in this case? – 2017-01-21

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I am just guessing.

Let $z=3i\pi$ and $w=3$. Note that $$ \log z = \ln |z|+i\theta= \ln(3\pi)+i\frac{\pi}{2}.$$

Since $\exp(3i\frac{\pi}{2})=-i$, then $$ \exp(3\ln(3\pi)+3i\frac{\pi}{2})) =-i(3\pi)^3=-27i\pi^3.$$ So, $\log(\exp(w\log(z))$ equals $$=\log(\exp(3\log(3i\pi))=\log(\exp(3\ln(3\pi)+3i\frac{\pi}{2}))=\log(-27i\pi^3)=\ln(27\pi^3)+i\frac{\pi}{2},$$ and $w\log z$ equals $$w\log z =3\log (3i \pi)=3(\ln(3\pi)+i\frac{\pi}{2})=3\ln(3\pi)+3i\frac{\pi}{2}.$$ Thus, $\log(\exp(w\log(z)) \neq w\log z.$