# Complex Logarithm

For what values of $p$ is the following valid? $$\log(z^p) = p\log(z)$$ where $$\log(z) = \ln(|z|) + i[\arg(z)+2\pi n]$$ where $n$ is an integer.

I heard the expression above should not be valid for all p so I computed $\log((1+i)^n)$ and $\frac1n \log(1+i)$ for $n = \frac12$ and $2$, but got the same answer so I know I've done something incorrectly.

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Your expression for $\log(z)$ is not a function. You must choose a particular value for $n$. Then you can see where the problem lies. –  ncmathsadist Feb 7 '13 at 2:30
As already noted, it depends on what you mean by ''log''. Is it a particular branch (which one?) or a multi-valued expression? –  mrf Feb 7 '13 at 9:39