Q.Evaluate $\log_{8}32$ using the change-of-base rule.

Re-expressing both 8 and 32 as powers of 2 lets the logarithm be evaluated through the common base 2.

By the change-of-base rule $\log_{a}N = \dfrac{\log_{b}N}{\log_{b}a}$, choose the convenient common base $b = 2$ (since both 8 and 32 are powers of 2):

$\log_{8}32 = \dfrac{\log_{2}32}{\log_{2}8}.$ …

Cross-check via base 10. $\log_{8}32 = \dfrac{\log_{10}32}{\log_{10}8} = \dfrac{5\log 2}{3\log 2} = \dfrac{5\times0.3010}{3\times0.3010} = \dfrac{1.5050}{0.9030} = 1.6667$. The $\log 2$ factor cancels, giving th …

Students sometimes invert the ratio and write $\dfrac{\log_{2}8}{\log_{2}32} = \dfrac{3}{5}$. In the change-of-base rule the logarithm of the number (32) goes on top a …