 # How do I simplify \log (1/sqrt(1000)) tidones0r 2022-09-14 Answered
How do I simplify $\mathrm{log}\left(1/\sqrt{1000}\right)$?
What I have done so far:
1) Used the difference property of logarithms
$\mathrm{log}\left(\frac{1}{\sqrt{1000}}\right)=\mathrm{log}\left(1\right)-\mathrm{log}\left(\sqrt{1000}\right)$
2) Used the exponent rule for logarithm
$\mathrm{log}\left(1\right)-\frac{1}{2}\mathrm{log}\left(1000\right)$
I'm stuck at this point. Can someone explain why and what I must do to solve this equation?
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Hint:
$\frac{1}{\sqrt{1000}}={10}^{-\frac{3}{2}}\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\mathrm{log}{x}^{a}=a\mathrm{log}x$
###### Not exactly what you’re looking for? Marcus Bass
${\mathrm{log}}_{10}\left(\frac{1}{\sqrt{1000}}\right)={\mathrm{log}}_{10}\left(\frac{1}{\sqrt{{10}^{3}}}\right)={\mathrm{log}}_{10}\left(\frac{1}{{10}^{\frac{3}{2}}}\right)={\mathrm{log}}_{10}\left({10}^{\frac{-3}{2}}\right)=\frac{-3}{2}$