Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

How can i simplify this:

$\dfrac{\log_2 625}{\log_2 125}$


share|cite|improve this question
Please reformat your question. – mixedmath May 2 '11 at 7:07
reformat already.. – dramasea May 2 '11 at 7:11
You can use $log(a\cdot b)=log(a)+log(b)$ to solve this. – Rasmus May 2 '11 at 7:14
can do it?? i dont cant get it – dramasea May 2 '11 at 7:18
Or even $\log(a^n)=n\log(a)$. – Phira May 2 '11 at 7:19
up vote 10 down vote accepted

Here it is:

$$\frac{\log_{2}625}{\log_{2}125} = \log_{125}625 = \frac{\log_{5}625}{\log_{5}125} = \frac{\log_{5}5^4}{\log_{5}5^3} = \frac{4}{3}$$

by virtue of a basic property of logarithms. The basic property I am referring to is the following:

$$ \frac{\log_{a}x}{\log_{a}y} = \log_{y}x $$

share|cite|improve this answer
I'm asking $\frac{log_2 625}{log_2 125}$ – dramasea May 2 '11 at 7:27
@dramasea: That's what he answered - he references the property that allows him to do that. – mixedmath May 2 '11 at 7:30
@dramasea: Actually I use it twice: the first time, I use $\frac{log_{a}x}{log_{a}y} = log_{y}x$, and the second time I use the 'reverse': $log_{y}x = \frac{log_{a}x}{log_{a}y}$. Do you know this equality? – Thomas Connor May 2 '11 at 7:33
It's called the Change of Base Property – Nicolas Villanueva May 2 '11 at 7:34
Thanks for detail explaination, Thomas. – dramasea May 2 '11 at 7:58

Without a change of base:

$$\frac{\log_2 625}{\log_2 125} =\frac{\log_2 \left( 5^4\right)}{\log_2 \left(5^3\right)} =\frac{4 \log_2 5}{3 \log_2 5} =\frac{4 }{3 } .$$

share|cite|improve this answer
@Henry: Consider to use hints. – AD. May 2 '11 at 10:35
This is the simplest of all answers. Using a change of base formula and then writing everything as $5^4$ and so on, instead of just writing it as $5^4$ in the first place only adds unnecessary steps. – Graphth May 2 '11 at 13:23
@AD.: There were already two answers (one accepted) giving $4/3$ when I replied – Henry May 2 '11 at 15:25
@Numth: I couldn't agree more. – Chris Leary May 2 '11 at 17:47
@Numth: I also agree, this is the most transparent answer - I gave the same comment on to the other writers too. – AD. May 2 '11 at 19:41

Although you have now changed your question, the spirit of Thomas's answer is still correct. In particular, $\dfrac{log_2 625}{log_2 125} = \dfrac{log_5 625}{log_5 125} = \dfrac{4}{3}$. Does that make sense?

share|cite|improve this answer
@Thomas: Aha! I see you too have now changed your answer to his question. Perhaps I will beat you to it before he changes it next. – mixedmath May 2 '11 at 7:23
;-) – Thomas Connor May 2 '11 at 7:25
Consider to use hints. – AD. May 2 '11 at 10:36

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.