# Logarithm Base Changing Formula Proof

In a logarithmic expression, it is possible to change base using algebraic manipulation.  For example, we can change

$log_416$ to $\dfrac{\log_216}{\log_24}$.

In this post, we are going to prove why it is possible to do such algebraic manipulation. The change of base above can be generalized as

$\log_ab = \dfrac{\log_cb}{\log_ca}$.

Theorem

$\log_ab = \dfrac{log_cb}{\log_ca}$.

Proof

If we let $\log_ab = x$, then by definition, $a^x = b$.

Now, take the logarithm to the base $c$ of both sides. That is

$log_c a^x = \log_cb$.

Simplifying the exponent, we have

$x \log_ca = \log_cb$.

Now, since $a \neq 1$, $\log_ca \neq 0$.

Therefore,

$x = \dfrac{\log_cb}{\log_ca}$

Thus,

$\log_ab = \dfrac{log_cb}{\log_ca}$