Reverse map for an equation .

Question

I don't know this is actually reverse mapping or what but i have following equation.

$$x = \tanh(a \cdot b ) + c $$

How do I solve for $a$?
Does it has anything to do with inverse hyperbolic function?

Answer 1

Yes, you can solve for $a$ by using the inverse hyperbolic tangent: $$x = \tanh ab + c$$ $$x-c = \tanh ab$$ $$\tanh^{-1}(x-c) = ab$$ $$a = \boxed{\dfrac1b \tanh^{-1}(x-c)}$$

Incidentally, the inverse hyperbolic tangent can be written in terms of perhaps more familiar functions as $$\tanh^{-1} y = \log\sqrt{\frac{1+y}{1-y}}.$$