Question 433406
{{{A = ln(x) - ln(e^(Bx)) = ln(x/(e^(Bx)))}}}


{{{e^A = x/e^(Bx)}}}


{{{e^A = xe^(-Bx)}}} Multiply both sides by -B


{{{-Be^A = -Bxe^(-Bx)}}}


There is a certain function called the Lambert W-function defined as the inverse of the function {{{f(z) = ze^z}}}. Here, we have {{{f(-Bx) = -Bxe^(-Bx)}}} so it follows that if we take the W-function of both sides,


{{{-Bx = W(-Be^A)}}}


{{{x = (-1/B)W(-Be^A)}}}


This is pretty much the simplest form that one can express x in, even though the W-function is somewhat obscure. There is, of course, using Newton's method, but it is very unlikely that a real number will be obtained.


Also, I checked my answer on Wolfram Alpha by putting arbitrary values for A and B (e.g. A = 253, B = 19), and it came out with the solution {{{x = (-1/19)W(-19e^253)}}}.


http://www.wolframalpha.com/input/?i=253+%3D+ln%28x%29+-+19x


Wikipedia article about the W-function:
http://en.wikipedia.org/wiki/Lambert_W_function