Showing the identity log($z^n$) = n log (z) for a particular value of n and z

I was asked to show $\log(z^n) = n \log (z)$ where $z = 1 + i$ and $n = 5$.

The worked solutions state that they are not equal for those values but I do not understand why given that we find the principal value for the first equation and not the second.

My working out is as follows:

For $n \log(z)$:

$5 \log(1+i) = 5 \log \sqrt2 +i\frac{5 \pi}{4} = 5 \log \sqrt2 +i\frac{-3 \pi}{4}$

On the other hand, for $\log(z^n)$:

$\log(\sqrt2 ^5 e^{i\frac{5\pi}{4}}) = \log(\sqrt2 ^5 e^{i\frac{-3\pi}{4}} ) = 5 \log \sqrt2 +i\frac{-3 \pi}{4}$

which means they are equal for the value of $n = 5$ and $z = 1+i$.

Can someone kindly explain what I am overlooking?