Logarithms via change of base

Text

Compute $\log_9 27$ and $\log_8 32$ by writing everything as powers of a common number.

Solution

First logarithm. $9=3^{2}$ and $27=3^{3}$, so by the change-of-base formula $\log_9 27=\frac{\log_3 27}{\log_3 9}=\frac{3}{2}=1.5.$ Second logarithm. $8=2^{3}$ and $32=2^{5}$, so $\log_8 32=\frac{\log_2 32}{\log_2 8}=\frac{5}{3}\approx 1.667.$ $\boxed{\,\log_9 27=\tfrac32,\quad \log_8 32=\tfrac53\approx 1.667\,}$