Express x in terms of constants.

Question

I have the expression,$ A(Bx + 1) = Cd^{2x}$ where A,B,C and d are constants. How to arrive at an expression for x in terms of A,B,C and d?

Answer 1

Your equation contains only elementary functions. An elementary function is a function which can be represented by applying only $\exp$, $\ln$ and arithmetic operations.

Convert your equation into that form ($d^{2x}=e^{(2x\ln(d))}$):

$A(Bx+1)=Ce^{(2x\ln(d))}$

The solution variable ($x$) is argument of functions which are algebraically independent. That means your equation has no solution that is an elementary function.

Solve your equation by the solution formula for equations of that type with help of Lambert W function (see e.g. Wikipedia: https://en.wikipedia.org/wiki/Lambert_W_function#Generalizations):

$$e^{-cx}=a(x-r)$$

$$x = r + \frac{1}{c}LambertW(\frac{ce^{-cr}}{a})$$