Analytical expression for the Logistic function mirrored in diagonal plane?

Question

I want to fit some data which is a normalised cumulative histogram which looks like so:

I was thinking that to fit this S shaped cumulative distribution I wanted a logistic function but if I attempt this I get the following:

This is with the expression $$ f(x) = \dfrac{L}{1 + e^{-k(x-x_0)}} $$ where $L = 1$, $k = 0.02$ and $x_0$ = 0.

I therefore need the mirror of the logistic function in the diagonal plane, is there an analytical function which is the diagonal mirror of the logistic function or do I need to mirror the logistic function using a geometric transformation?

Edit with solution:

caverac's answer below was the correct solution with the modification he proposed in the comments:

$$ g(x) = -a\log\left(\frac{k}{(x+c)} - 1\right) + b $$

with the following values:

• $a = 0.2$
• $b = 0.5$
• $c = 300$
• $k = 600$

This produces the following plot:

Where the red line is the equation above.

Answer 1

If you invert your relation you can get the "diagonal reflection" you're looking for

$$ g(x) = -a\log\left(\frac{1}{x} - 1\right) + b $$