# How do you convert a logarithm to a different base?

Oct 22, 2015

If you want to convert ${\log}_{a} \left(x\right)$ into ${\log}_{b} \left(x\right)$, use the following equality:
${\log}_{b} \left(x\right) = {\log}_{b} \left(a\right) \cdot {\log}_{a} \left(x\right)$

#### Explanation:

Consider $y = {\log}_{a} \left(x\right)$.
From a definition of logarithm
(1) $x = {a}^{y}$
Analogously, if $z = {\log}_{b} \left(x\right)$,
(2) $x = {b}^{z}$
Finally, if $r = {\log}_{b} \left(a\right)$,
(3) $a = {b}^{r}$

From (1) and (2) we derive
(4) ${a}^{y} = {b}^{z}$

Using (3) and substituting $a$ in (4) we get
(5) ${\left({b}^{r}\right)}^{y} = {b}^{z}$
or, simplifying the left part,
(6) ${b}^{r \cdot y} = {b}^{z}$

From the last equation we see that
$z = r \cdot y$
or, returning back to the meaning of $r$, $y$ and $z$,
${\log}_{b} \left(x\right) = {\log}_{b} \left(a\right) \cdot {\log}_{a} \left(x\right)$