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What is $\displaystyle \log \frac{1}{\sqrt{10}}$?

Thanks so much beforehand.

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2 Answers

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Firstly $\ln (1/\sqrt{10})=\ln (\sqrt{1/10})=\frac{-1}{2} \ln 10$. $\ln 10$ isn't a rational number, but rather $\ln 10 \approx 2.302585$. Thus, $\ln (1/\sqrt{10})=-1.1512925$.

For the chance you wanted $\log_{10}$, we have all the same as the above, except now $\log_{10}(10)=1$, so that $\log_{10}(1/\sqrt{10})=-1/2$.

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$$\log\frac ab=\log a-\log b\implies \log\frac1{\sqrt{10}}=\log1-\log\sqrt{10}=-\frac12\log 10\;\;\text{(why?)}$$

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