How do I simplify $\log (1/\sqrt{1000})$?
How do I simplify $\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right)$?
What I have done so far:
1) Used the difference property of logarithms $$\log \left(\displaystyle\frac{1}{\sqrt{1000}}\right) = \log(1) - \log(\sqrt{1000}) $$
2) Used the exponent rule for logarithm
$$\log (1) - \frac{1}{2}\log (1000) $$
I'm stuck at this point. Can someone explain why and what I must do to solve this equation?
$\endgroup$ 33 Answers
$\begingroup$Hint:$$\frac{1}{\sqrt{1000}}=10^{-\frac{3}{2}}\qquad\mbox{and}\qquad\log x^a=a\log x$$
$\endgroup$ 3 $\begingroup$$\log_{10} \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log_{10} \left( \displaystyle \frac{1}{\sqrt{10^3}} \right) = \log_{10} \left( \displaystyle \frac{1}{{10^\frac{3}{{2}}}} \right) = \log_{10} \left( \displaystyle {{10^\frac{-3}{{2}}}} \right) = \displaystyle {{\frac{-3}{{2}}}}$
$\endgroup$ $\begingroup$$\log_{10} \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log_{10} \left( \displaystyle \frac{1}{\sqrt{10^3}} \right) = \log_{10} \left( \displaystyle \frac{1}{{10^\frac{3}{{2}}}} \right) = \log_{10} \left( \displaystyle {{10^\frac{-3}{{2}}}} \right) = \displaystyle {{\frac{-3}{{2}}}}$
Sorry I accidently erased an answer. Please restore. I submit mine as another.
$\endgroup$ 4
