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

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

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

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

**Theorem**

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

**Proof**

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

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

$latex log_c a^x = \log_cb$.

Simplifying the exponent, we have

$latex x \log_ca = \log_cb$.

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

Therefore,

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

Thus,

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