Change of Base Formula

9. Change of base formula. For $a,x,b>0$ and $a,b\ne1$: $\log_ax=\dfrac{\log_bx}{\log_ba}$ — a logarithm in one base can always be rewritten as a ratio of logarithms in any other convenient base $b$.

Note: as a special case, $\log_ax=\dfrac{\log_xx}{\log_xa}=\dfrac{1}{\log_xa}$ (verify!) — swapping the base and the argument turns a logarithm into its own reciprocal.

Ex. 16: Evaluate $\dfrac{\log_481}{\log_49}$.

Solution: By the change of base law, since the base is the same (namely 4) in both the numerator and denominator, $\dfrac{\log_481}{\log_49}=\log_981=2$ (since $9^2=81$).

Ex. 17: Prove that $2\log_ba^4\cdot\log_cb^3\cdot\log_ac^5=120$. …