How to solve logarithms equations

Question

I've some problems with solving of those logarithms equations:

1) $\ln(x+1)\le x$

2) $x^2-8\ln(x)=0$

Answer 1

For 1 consider the function $$ f(x)=\ln(x+1)-x $$ defined over $(-1,\infty)$, whose derivative is $$ f'(x)=\frac{1}{x+1}-1=-\frac{x}{x+1} $$ Can you find the absolute maximum?

For 2 consider the function $$ g(x)=x^2-8\ln x $$ defined over $(0,\infty)$, whose derivative is $$ g'(x)=2x-\frac{8}{x}=\frac{2(x^2-4)}{x} $$ Can you find the absolute minimum? What can you deduce?