How do I evaluate this logarithm expression: $10 \log_{10} (x/y) = -20$?

Question

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.

The answer $0.01$, or equivalently $10^{-2}$, is correct. – André Nicolas Feb 13 '14 at 04:56 — André Nicolas, Feb 13 '14 at 04:56
it seems that we have $log_{10}(x/y)$ right – dato datuashvili Feb 13 '14 at 05:00 — dato datuashvili, Feb 13 '14 at 05:00

score 5 · Accepted Answer · answered Feb 13 '14 at 04:57

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 $$ \begin{split} e^{\log z} &= e^{-2} \\ z &= e^{-2} \end{split} $$

score 3 · Answer 2 · answered Feb 13 '14 at 04:57

3

divide both members by 10.

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

--> 10^-2 = x/y

--> 0.01 = x/y

answered Feb 13 '14 at 04:57

Diophanties

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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 04:58
your answer is correct my friend +1 – dato datuashvili Feb 13 '14 at 05:01

score 1 · Answer 3 · answered Feb 13 '14 at 04:59

1

ok we have following thing

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

or

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

answered Feb 13 '14 at 04:59

dato datuashvili

9,194

How do I evaluate this logarithm expression: $10 \log_{10} (x/y) = -20$?

3 Answers3