Seeking solution for: $16^x+36^x=81^x$

Question

Solve for x:

$$16^x+36^x=81^x$$

This question was given by one of the student in the lesson. Can this question be solved?

A possible approach is to take the log

$$x\log(81)=\log(16^x+36^x)$$

It is impossible to isolate the $x$ on the RHS.

Answer 1

We can rewrite it as

$$\left(\frac{16}{81}\right)^x+\left(\frac{36}{81}\right)^x=1 \implies\left(\frac{4}{9}\right)^{2x}+\left(\frac{4}{9}\right)^x-1=0$$

Let $y=\left(\frac{4}{9}\right)^x$.

Solve the quadratic in $y$, then substitute $x = \frac{\ln y}{\ln(\frac49)}$