Answer
$x \approx 0.76$
Work Step by Step
RECALL:
$\ln{a^b} = b \cdot \ln{a}$
Take the natural logarithm of both sides to obtain:
$\ln{2^{x+1}}=\ln{5^x}$
Use the rule above to obtain:
$(x+1)\ln{2}=x\ln{5}$
Distribute $\ln{2}$ to obtain:
$x\ln{2} + \ln{2} = x\ln5$
Subtract $\ln{2}$ on both sides to obtain:
$x\ln{2} = x\ln{5}-\ln{2}$
Subtract $x\ln{5}$ on both sides of the equation to obtain:
$x\ln{2} - x\ln{5}=-\ln{2}$
Factor out $x$ on the left side of the equation to obtain:
$x(\ln2-\ln5)=-\ln{2}$
Divide $(\ln{2} - \ln{5})$ on both sides of the equation to obtain:
$x = \dfrac{-\ln{2}}{\ln{2}-\ln{5}}$
Use a scientific calculator to obtain:
$x= 0.7564707974
\\x \approx 0.76$