Don't know how to go on for algebra problem

Question

This is a problem that I stumbled upon: $\ln\frac{(x+5)}{(x-4)}=x$. It seemed relatively simple at first, but I ended up at $(e^x-1)(x-4)=9$. I can't really figure out how to continue from there. I feel like I'm missing something obvious, but any help would be appreciated.

Answer 1

Equations with general functions, like logarithms, cosines, and others, aren't necessarily going to have nice integer, or even rational, answers. They might not even have answers with a nice algebraic expression, like $\sqrt{2}$ or something. This is one such case. In particular, you can tell that if $x$ is an integer then $\ln\left(\frac{x+5}{x-4}\right)$ is an integer and that only happens for rational inputs to the natural log when the argument is 1. $\frac{x+5}{x-4}=1$ has no solution so this equation is not going to have any nice solution.

If you wanted to try to solve this algebraically you might try exponentiating both sides by $e$ and distributing, then trying to group all terms with an $x$ in them. That would result in

$$\frac{x+5}{x-4}=e^x$$

$$x+5= xe^x-4e^x$$

$$5 = xe^x-4e^x -x$$

From here there's no obvious algebraic next step since this doesn't factor in the usual ways. So algebraically this looks like a dead end.

I'd sooner believe that the author of the problem expects you to graph $\ln\left(\frac{x+5}{x-4}\right)$ on the one hand, and graph $x$ on the other, and using a graphing calculator, find the approximate intersection of these two.