# 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?

• $\log 10000 = \log 10^4 = 4 \log 10$ Jun 28, 2014 at 18:35
• $\log 1=0$ and $\log 10000=4\log 10$ so this become $-2\log 10$.
– Gina
Jun 28, 2014 at 18:36
• Looks like you added an extra $0$ to the $1000$ after the step (2).
– Cure
Jun 28, 2014 at 18:36

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

• Is the answwer -3/2log(10)? Jun 28, 2014 at 19:48
• @Prologue: Yes! Jun 28, 2014 at 19:50
• @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. Jun 28, 2014 at 19:59

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

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

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

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