Equation for length L in terms of y-length

OB = 1
OA = L

Equation to find the length L if y-length of BA, relative to O, is given.

The length of L equals e raised to the power of y-length times 2, plus 1 : divided by 2 times e raised to the power of y-length.

L = (e^2ylength + 1) / 2e^ylength

in the diagram:
AB = 1.959592
y-length (of BA, relative to O) = 1.425417

the distance L (AO) is:

L = (e^2(1.425417) + 1) / 2e^1.425417
= 18.302206 / 8.319184
= 2.2