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How can i simplify this:

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

Thanks

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    @dramasea: please don't do what you did on this question again. The latter edits you made completely changed the meaning/character of the question, which would've invalidated other users' helpful answers.2011-05-02

3 Answers 3

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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?

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    Consider to use hints.2011-05-02
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Without a change of base:

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

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    @Henry: Your answer is the best one - however I am sure you can write that answer in hints..2011-05-02
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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 $

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    @Thomas Connor: Consider to use hints.2011-05-02