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}

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