Answer
(a) $x=\frac{1}{5}\log\frac{5}{4}$
(b) $x\approx 0.019382$
Work Step by Step
(a) We need to solve:
$4(1+10^{5x})=9$
$1+10^{5x}=\frac{9}{4}$
$10^{5x}=\frac{9}{4}-1$
$10^{5x}=\frac{5}{4}$
We take the log of both sides and use log rules to simplify:
$\log 10^{5x}=\log \frac{5}{4}$
$5x=\log\frac{5}{4}$
$x=\frac{1}{5}\log\frac{5}{4}$
(b) We solve using a calculator:
$x\approx 0.019382$
