Answer
$x=\dfrac{\log\dfrac{50}{3}}{\log4}\approx2.029447$
Work Step by Step
$4^{x}+2^{1+2x}=50$
We can rewrite the expression on the left side of the equation using the product of powers of the same base rule:
$4^{x}+(2)(2^{2x})=50$
Apply the power of a power rule to the second term of the left side of the equation:
$4^{x}+(2)(2^{2})^{x}=50$
$4^{x}+(2)(4^{x})=50$
Simplify and solve for $4^{x}$:
$3(4^{x})=50$
$4^{x}=\dfrac{50}{3}$
Apply $\log$ to both sides of the equation:
$\log4^{x}=\log\dfrac{50}{3}$
The exponent $x$ can be taken down to multiply in front of the $\ln$:
$x\log4=\log\dfrac{50}{3}$
Solve for $x$:
$x=\dfrac{\log\dfrac{50}{3}}{\log4}\approx2.029447$