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

Apr 5, 2018

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

#### Explanation:

I am assuming by common logs this means base 10.

If:

$y = {\log}_{b} a \iff {b}^{y} = a$

Suppose we wish to express this using a different base. Let's say to a base $\boldsymbol{c}$.

From:

${b}^{y} = a$

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

$y {\log}_{c} b = {\log}_{c} a$

Divide by ${\log}_{c} b$:

$y = \frac{{\log}_{c} a}{{\log}_{c} b}$

From above:

$y = {\log}_{b} a$

$\therefore$

${\log}_{b} a = \frac{{\log}_{c} a}{{\log}_{c} b}$

This is the change of base formula:

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