Proving logarithmic inequality

Hello, everyone!

I'm stuck at the following problem.

Let us have an equation that I intend to solve with some numerical method:

, where is a small number and is positive.

First off, I want to get rid of the modulus sign to avoid solving a system of equations. To do so, I need to show that

(This fact I established empirically and want to prove. The plot can be seen here: 1 is actually a limit).

The inequality is equivalent to the following one:

Denote ( ), then

And here's where I'm out of ideas. How do I analytically prove the last inequality for all ?

Any thoughts would be very appreciated!

PS I probably should have posted this in Calculus subforum, my bad.