I'm looking for an example of a transcendental equation with a unique "nice" solution that can be identified as correct by inspection. My first thought was $$3^x = x^3$$ but this equation has two solutions, one of which is trivial and the other of which requires $W$-functions.
However, it seems to me that if I introduce a constant factor on one side of the equation, I ought to be able to get the two function $f(x)=3^x$ and $g(x)=kx^3$ to be tangent to each other at a single point of intersection. More broadly, it seems like if $a$ and $b$ are any positive numbers (and $b$ is odd) there should be some constant $k$ such that $a^x = kx^b$ has a unique positive solution.
I doubt there's a simple, closed form solution for this general problem that doesn't involve $W$ functions, but I don't really care about the general case; I just want one clear example. So the question:
Can you provide a specific example of an equation of the form $x^a = k\cdot b^x$ for which $a, b, k$ are reasonably "nice", and the equation has a unique solution that's also reasonably "nice"? Alternatively, can you confirm that no such example exists?