Two logarithms with change of base

Statement

Using the change-of-base formula and natural logarithms, compute to three decimal places:

• (a) $\log_8 20$
• (b) $\log_9 5$

Solution

Formula: $\log_b a=\dfrac{\ln a}{\ln b}$.

(a) $\log_8 20=\dfrac{\ln 20}{\ln 8}=\dfrac{2{.}99573}{2{.}07944}\approx 1{.}441$.

(b) $\log_9 5=\dfrac{\ln 5}{\ln 9}=\dfrac{1{.}60944}{2{.}19722}\approx 0{.}732$.

$\boxed{\log_8 20\approx 1{.}441;\qquad \log_9 5\approx 0{.}732}$