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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?

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    No. I have a special application my calculator through which I am able to make logical expressions. Like 5 = 5 will return true on that app. My calculator is giving me false for this logic.2012-06-24

1 Answers 1

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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?

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    Sorry. I had made I mistake. I was comparing my calculator with WA's calculator and it slipped by mind that default logarithm WA uses is the natural logarithm. I was using the log with a base of 10 instead of with a base of $e$2012-08-03