On the equation $\exp(a x+b)=\ln(x)$

Question

I am confronted with:

$$\exp(a x+b)=\ln(x)$$

for $a,b$ reals and $a<0$, $b>0$.

I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of elementary functions (perhaps augmented a bit e.g. with Lambert's W function ). Otherwise I will have to work out some approximation scheme.

Regarding the analytic solution, my research has not revealed anything until now so any pointer for further study (that I might have missed) is more than welcome.

On the approximation case, due to the mixing of the exponential and the logarithm, things are not obvious. So any opinion or clue would be very useful.

Answer 1

I find it highly doubtful that you will find an analytic solution in terms of elementary functions.

Given your constraints, the solution is somewhere in $(1,\infty)$. To me, it seems best to take the log of both sides and find the root of $$ f(x) = \ln\ln x - ax-b$$ If all you need is a computational solution, then applying Newton's method to a starting guess of $x_0=1+\epsilon$ should get a solution in a few iterations.

Here is some mathematica code for the root as a function of $a$ and $b$:

f[a_,b_] := x /. FindRoot[Log[Log[x]] - (ax+b), {x,1+1*^-15}];

Plotting this gives:

With some more effort, I'm sure a reasonable approximation to this surface can be found.