Solve $4 \times2^x+3^x=5^x$ without any sort of calculator Is there a way i can solve the following equation only by using high school mathematics?
$$4 \times2^x+3^x=5^x$$
I tried writing $5$ as $2+3$ but didn't get any result.
After that i tried to divide by $5^x$ and see how the function goes, but, unfortunately didn't got me somewhere either.

Assuming $x$ to be natural,
$$
5^x-3^x=(5-3)(5^{x-1}+3\cdot5^{x-2}+3^2\cdot5^{x-3}+\cdots+3^{x-1})=4\cdot2^x\\
\implies (5^{x-1}+3\cdot5^{x-2}+3^2\cdot5^{x-3}+\cdots+3^{x-1})=4\cdot2^{x-1}
$$
Each term in LHS looks like $3^{n}5^{x-1-n}>2^n2^{x-1-n}=2^{x-1}$.
There are $x$ number of terms in LHS. If there are $4$ or more terms in LHS, each of them would be greater than $2^{x-1}$ and the equation would not hold. So $1\le x\le 3$. Manually checking $x=1,2,3$ we see $x=2$ is a solution.
