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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?

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    $\begingroup$ $\log 10000 = \log 10^4 = 4 \log 10$ $\endgroup$ – vonbrand Jun 28 '14 at 18:35
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    $\begingroup$ $\log 1=0$ and $\log 10000=4\log 10$ so this become $-2\log 10$. $\endgroup$ – Gina Jun 28 '14 at 18:36
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    $\begingroup$ Looks like you added an extra $0$ to the $1000$ after the step (2). $\endgroup$ – Cure Jun 28 '14 at 18:36
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Hint: $$\frac{1}{\sqrt{1000}}=10^{-\frac{3}{2}}\qquad\mbox{and}\qquad\log x^a=a\log x$$

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    $\begingroup$ Is the answwer -3/2log(10)? $\endgroup$ – Prologue Jun 28 '14 at 19:48
  • $\begingroup$ @Prologue: Yes! $\endgroup$ – TonyK Jun 28 '14 at 19:50
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    $\begingroup$ @Prologue Note that, in base 10, this simplifies to $-\frac{3}{2}.$ If you're in base $e$, then ignore the first part of this sentence. $\endgroup$ – beep-boop Jun 28 '14 at 19:59
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$\log \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log1 - \log\sqrt{1000}$

$\implies\log1-\frac{1}{2}\log10^3$

$\implies\log1-\frac{3}{2}\log10$

Now, $\log1=0$ as $10^0=1$

and $\log10=1$ as $10^1=10$

Substituting the values gives us the answer as $-\frac{3}{2}$.

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