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If $\frac{log(a)}{log(b)}=1000,$ then what is the numerical value of $\frac{log(a/b)}{log(b)}$?

I was not sure on how to solve this. I was just looking at this problem to see how would you solve it. I tried doing this but I don't see how you can get a numerical value from this compared to other problems:

$\frac{log(b)*1000}{log(a)}$

Can someone please help me with this solution?

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  • $\begingroup$ Do you know any ways to rewrite $\log(a/b)$? If not, what about $\log(ab)$? $\endgroup$
    – Kevin Arlin
    Sep 15, 2014 at 4:06

2 Answers 2

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Use the laws of logs.

$\log(\frac ab) = \log a - \log b$

So the expression becomes $\frac{\log a - \log b}{\log b} = \frac{\log a}{\log b} - 1$ which equals $999$.

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  • $\begingroup$ Thank you. I thought about doing it that way but ended up cross multiplying and got confused with the $log(a/b)$ in the parenthesis compared to log(a)/log(b). Thank you. $\endgroup$
    – col
    Sep 15, 2014 at 4:26
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Use the fact that $\log(a/b)=\log a-\log b$. Then

$$\frac{\log(a/b)}{\log b}=\frac{\log(a)}{\log b}-1.$$

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