Answer
(a) $x=\frac{\ln\frac{50}{3}}{\ln 4}$
(b) $x\approx 2.029447$
Work Step by Step
(a) We need to solve:
$4^{x}+2^{1+2x}=50$
$(2^2)^x+2^{1+2x}=50$
$2^{2x}(1+2^1)=50$
$2^{2x}(3)=50$
$2^{2x}=\frac{50}{3}$
We take the log of both sides and use log rules to simplify (we choose the natural log, but any other base would work as well):
$x \ln2^2 = \ln 50-\ln 3=\ln\frac{50}{3}$
$x=\frac{\ln\frac{50}{3}}{\ln 4}$
(b) We solve using a calculator:
$x\approx 2.029447$
