# How do I evaluate this logarithm expression?

I think the answer for this is 0.01, but I'm not sure. Could someone explain the steps in solving the following for $(x/y)$:

$$10 \log_{10} (x/y) = -20$$

I've tried putting $\frac{-20}{10 \log_{10}}$ in Wolfram Alpha, but the answer doesn't look like what I was expecting.

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The answer $0.01$, or equivalently $10^{-2}$, is correct. – André Nicolas Feb 13 '14 at 4:56
it seems that we have $log_{10}(x/y)$ right – dato datuashvili Feb 13 '14 at 5:00

Let $z=x/y$ so you have $$\begin{split} 10 \log z &= -20 \\ \log z &= -2 \\ 10^{\log z} &= 10^{-2} \\ z &= 0.01 \end{split}$$ assuming your log was base 10. If it was base $e$, the last 2 steps are $$\begin{split} e^{\log z} &= e^{-2} \\ z &= e^{-2} \end{split}$$

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+1 for nice answer – dato datuashvili Feb 13 '14 at 5:03

divide both members by 10.

--> log (x/y) = -2

--> 10^-2 = x/y

--> 0.01 = x/y

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oops, was that second ten a base on the logarithm or a coefficient of (x/y)? – Diophanties Feb 13 '14 at 4:58
your answer is correct my friend +1 – dato datuashvili Feb 13 '14 at 5:01

ok we have following thing

$log_{10}(x/y)=-2$

or

$(x/y)=10^{-2}$

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