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

to .

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

.

**Theorem**

.

**Proof**

If we let , then by definition, .

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

.

Simplifying the exponent, we have

.

Now, since , .

Therefore,

Thus,