Answer
(a) $x=\frac{\ln\frac{100}{3}}{3\ln 5}$
(b) $x\approx 0.726249$
Work Step by Step
(a) We need to solve:
$125^{x}+5^{3x+1}=200 $
$(5^3)^x+5^{3x+1}=200$
$5^{3x}(1+5^1)=200$
$5^{3x}(6)=200$
$5^{3x}=\frac{200}{6}=\frac{100}{3}$
We take the log of both sides and use log rules to simplify:
$\ln 5^{3x}=\ln \frac{100}{3}$
$3x\ln 5=\ln 100-\ln 3=\ln \frac{100}{3}$
$x=\frac{\ln\frac{100}{3}}{3\ln 5}$
(b) We solve using a calculator:
$x\approx 0.726249$
