Equation for inverted mirrored $\ln(x)$ for $x<1$?

Question

Is there a simple equation for this curve? It uses $\ln(x)$ model for $x>1$ (I removed the $\ln$ part below zero) and something like inverted/mirrored $\ln(x)$ for $x<1$. It's not $\arctan(x)$ because it scales differently.

Answer 1

If you want that exact function, you should write it as $$ f(x) = \cases{\ln(x) & if $x>1$\\-\ln(2-x) & otherwise} $$ For $x>1$ we have $\ln$, and for all the other values of $x$, we have mirrored it horizontally (as we can see from the $-x$), and mirrored it vertically (as we can see from the $-\ln$). (The $2$ in $2-x$ adjusts the axis of the horizontal reflection, so that we reflect about $x = 1$, not $x = 0$.)

Answer 2

The mirrored part is

$$-\ln (2-x)$$

Answer 3

Another representation: $$ \log\left(\frac{(x-1)_+^2-1}{x-2}\right) $$ where $(\dots)_+$ is the positive part function: $$ (x)_+=\frac{|x|+x}2 $$