Exponential equation solved with logarithms

Statement

Solve the following exponential equations using logarithms, giving the result to two decimal places:

• (a) $1{.}045^{\,x}=3$
• (b) $0{.}85^{\,x}=0{.}2$

Solution

Take natural logarithms of both sides.

(a) $x\ln 1{.}045=\ln 3 \Rightarrow x=\dfrac{\ln 3}{\ln 1{.}045}=\dfrac{1{.}09861}{0{.}04401}\approx 24{.}96$.

(b) $x\ln 0{.}85=\ln 0{.}2 \Rightarrow x=\dfrac{\ln 0{.}2}{\ln 0{.}85}=\dfrac{-1{.}60944}{-0{.}16252}\approx 9{.}90$.

$\boxed{x_a\approx 24{.}96;\qquad x_b\approx 9{.}90}$