Answer
$x=\dfrac{\log\dfrac{100}{3}}{\log125}\approx0.726249$
Work Step by Step
$125^{x}+5^{3x+1}=200$
Use the product of powers rule to rewrite the left side of the equation like this:
$125^{x}+(5^{3x})(5)=200$
Now, use the power of a power rule to rewrite $5^{3x}$ like this:
$125^{x}+(5)(5^{3})^{x}=200$
$125^{x}+(5)(125^{x})=200$
Solve for $125^{x}$
$(6)125^{x}=200$
$125^{x}=\dfrac{200}{6}$
$125^{x}=\dfrac{100}{3}$
Apply $\log$ to both sides of the equation:
$\log125^{x}=\log\dfrac{100}{3}$
The exponent $x$ can be taken down t multiply in front of the $\log$:
$x\log125=\log\dfrac{100}{3}$
Solve for $x$:
$x=\dfrac{\log\dfrac{100}{3}}{\log125}\approx0.726249$