Solve logarithmic equation $2^x = 1 + 15\log_{5}(x+1)$

Question

$\ 2^x = 1 + 15\log_{5}(x+1)$

Is there any other way of solving this equation, except graphical?

Answer 1

Since $f(x) =2^x -1-15\log_{5}(x+1)$, $\ x>-1$ is a strictly convex function ($f''>0$), $y=f(x)$ and $y=0$ intersect at most two points. Since $f$ has $x=0$ and $x=4$ as its zeros, these are the only solutions to the equation.