Evaluate $\text{Ln}[(1+i)^7]$ The solution i obtain is $\text{Ln}[(1+i)^7]=7\ln2^0.5 + i(7\pi/4)$. Wolfram Alpha gives me a completely different answer and was hoping someone could either confirm this answer or tell me where i went wrong.
Thanks.
 A: Your answer is almost correct. Using the properties of logarithms, the first term can also be written $3.5\ln 2$. More importantly, the capital L in Ln indicates "principal value" which for logarithms has the argument in $(-\pi, \pi]$ so it should be $-i(\pi/4)$.
What they show on Wolfram Alpha is exactly $3.5 \ln 2 - i(\pi/4)$, except that they use the numerical values of $\ln 2$ and $\pi$.
A: Switchin to exp form:
$$1+i=\sqrt2e^{i\frac\pi4}\implies(1+i)^7=(\sqrt2)^7e^{i\frac74\pi}=8\sqrt2e^{i\frac74\pi}.$$
The log is then:
$$\log[(1+i)^7]=\log(8\sqrt2)-i\frac\pi4,$$
Assuming you use the prinipal log. First term will be $\frac72\log2=3.5\log2$. To have $i\frac74\pi$ you'll need to take a log that is not the principal log, easiest choice being the one defined on all complex numbers save for positives, which is not standard AFAIK. Note that one of Wolfram's alternate forms is exactly my result. You have (modulo fixing the exponent which is $^{0.5}$ and not $^0.5$, I take it $7\log2^{\frac12}$, which by properties of the real log is $\frac72\log2$. So fix the argument to $-\frac\pi4$ and Wolfram agrees with you :).
