Find x with $4^{x-1} = 9\cdot x^{3-x}+7$

Question

How to solve this exponential functuion?

$4^{x-1} = 9\cdot x^{3-x}+7$

The solution is $x=3$.

Answer 1

Equations such as this very rarely have closed-form solutions, and there is no general method for finding them. In this case it's easy to verify that $x=3$ works. Using the fact that for $x > 2$, the right side is decreasing while the left side is increasing, it's not hard to show this is the only positive real solution.