Answer
$x=\dfrac{\log4}{\log5-\log4}\approx6.212567$
Work Step by Step
$5^{x}=4^{x+1}$
Apply $\log$ to both sides of the equation:
$\log5^{x}=\log4^{x+1}$
The exponents $x$ and $x+1$ can be taken down to multiply in front of their respective logarithms:
$x\log5=(x+1)\log4$
Solve for $x$:
$x\log5=x\log4+\log4$
$x\log5-x\log4=\log4$
$x(\log5-\log4)=\log4$
$x=\dfrac{\log4}{\log5-\log4}\approx6.212567$