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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, 2014 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, 2014 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, 2014 at 18:36

3 Answers 3

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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, 2014 at 19:48
  • $\begingroup$ @Prologue: Yes! $\endgroup$
    – TonyK
    Jun 28, 2014 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, 2014 at 19:59
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$\log_{10} \left( \displaystyle \frac{1}{\sqrt{1000}} \right) = \log_{10} \left( \displaystyle \frac{1}{\sqrt{10^3}} \right) = \log_{10} \left( \displaystyle \frac{1}{{10^\frac{3}{{2}}}} \right) = \log_{10} \left( \displaystyle {{10^\frac{-3}{{2}}}} \right) = \displaystyle {{\frac{-3}{{2}}}}$

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

Sorry I accidently erased an answer. Please restore. I submit mine as another.

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  • $\begingroup$ Whose answer did you erase $\endgroup$
    – Some Guy
    Mar 24, 2021 at 22:21
  • $\begingroup$ MonK and maybe someone else $\endgroup$
    – Kav
    Mar 24, 2021 at 23:35
  • $\begingroup$ why would you erase dude? $\endgroup$
    – MonK
    Apr 16, 2021 at 12:18
  • $\begingroup$ because I might be in a rush when reading this question. $\endgroup$
    – Kav
    Apr 16, 2021 at 17:09

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