Answer
$x=\sqrt{10^5}-1\approx 315.2278$
Work Step by Step
Divide $2$ to both sides:
$\dfrac{2\log{(x+1)}}{2}=\dfrac{5}{2}$
$\log{(x+1)}=\frac{5}{2}$
Recall the definition of a logarithm (pg. 451):
$\log_{b}{x}=y$ iff $b^y=x$
Applying this definition to our equation (with $b=10, y=5/2, x=x+1$), we get:
$10^{\frac{5}{2}}=x+1$
$\left(10^5\right)^{1/2}=x+1$
$\sqrt{10^5}=x+1$
$x=\sqrt{10^5}-1$
$x=10^2\sqrt{10}-1\approx 315.2278$
We confirm that the answer works:
$2\log_{10}{(315.2278+1)}=5$
$2\log_{10}{(316.2278)}=5$
$2\cdot 2.5=5$
$5=5$
