Why does my calculator return false when I input $\log_{5}{-3} = \frac{\log(3)+\pi i}{\log(5)}$ but W|A returns true?
I'm thinking my calculator is wrong because I know that $\displaystyle \log_{5}{(-3)} = \frac{\log(-3)}{\log(5)}$ and $\displaystyle \log(-x) = \log(x) + \pi i$ so that means that $\displaystyle \log_{5}{(-3)} =\frac{\log(3) + \pi i}{\log(5)}$.
So why does my calculator return false?
Complex logarithms are multivalued, since the exponential is periodic with period $2 \pi i$. Thus, properly speaking, $$ \log_5(-3) = \frac{\log(3) + \pi i + 2 n \pi i}{\log(5) + 2 k \pi i} n,k \in \mathbb{Z} $$
So, your calculator may be using different values of $n$ and $k$ from your $n=k=0$. In the vocabulary of complex analysis, we might say that your calculator is returning a value from a different branch than you expect.
Which calculator are you working with?