How do I simplify $\log (1/\sqrt{1000})$?

Question

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?

Answer 1

Hint: $$\frac{1}{\sqrt{1000}}=10^{-\frac{3}{2}}\qquad\mbox{and}\qquad\log x^a=a\log x$$

Answer 2

$\log_{10} \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log_{10} \left( \displaystyle \frac{1}{\sqrt{10^3}} \right) = \log_{10} \left( \displaystyle \frac{1}{{10^\frac{3}{{2}}}} \right) = \log_{10} \left( \displaystyle {{10^\frac{-3}{{2}}}} \right) = \displaystyle {{\frac{-3}{{2}}}}$

Answer 3

$\log_{10} \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log_{10} \left( \displaystyle \frac{1}{\sqrt{10^3}} \right) = \log_{10} \left( \displaystyle \frac{1}{{10^\frac{3}{{2}}}} \right) = \log_{10} \left( \displaystyle {{10^\frac{-3}{{2}}}} \right) = \displaystyle {{\frac{-3}{{2}}}}$

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