Proof of the change of base

The formula is obtained by applying an auxiliary logarithm to both sides of the definition.

Proof — Change-of-base formula

Setting $u = \log_a x$, then $a^u = x$. We apply to both sides the logarithm in base $b$: $\log_b(a^u) = \log_b x \implies u \log_b a = \log_b x \implies u = \frac{\log_b x}{\log_b a}.$ $\blacksquare$

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Topics: Logarithmic function
Concepts: Change of base · Logarithm
Functions: Logarithmic function
Skills: Prove