How do you express $log_2 5$ in terms of common logs?

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Answer 1 · 2018-04-05T07:16:54

$color(blue)((log_(10)5)/(log_(10)2)$

I am assuming by common logs this means base 10.

If:

$y=log_(b)a<=>b^y=a$

Suppose we wish to express this using a different base. Let's say to a base $bbc$ .

From:

$b^y=a$

Take logarithms to the base $c$ of both sides:

$ylog_(c)b=log_(c)a$

Divide by $log_(c)b$ :

$y=(log_(c)a)/(log_(c)b)$

From above:

$y=log_(b)a$

$:.$

$log_(b)a=(log_(c)a)/(log_(c)b)$

This is the change of base formula:

$log_(2)5=(log_(10)5)/(log_(10)2$

How do you express log_2 5log_2 5 in terms of common logs?