How to solve this equation $8 \left(3^x+5^x+7^x\right)=5\cdot 2^x+2\cdot 4^x+17\cdot 6^x$?

Question

I use Mathematica to solve this equation $$8 \left(3^x+5^x+7^x\right)=5\cdot 2^x+2\cdot 4^x+17\cdot 6^x$$ and get three solutions $x=0\lor x=1\lor x=2.$ I don't know how to solve by hand. How can I solve it?

Answer 1

I'll assume you're searching solutions in $\mathbb{Z}$. Now it'easy to see that if $x$ is negative, let's say $x=-q$ with $q$ positive, the equation is not solvable as we can multiply both sides by $3^q5^q7^q$, so that the left side is an integer, but the other one clearly is not. On the other hand if $x$ is greater than $3$, we get that the greatest power of 2 dividing the left side is always $8$, whereas the right one is divisible by $2^x$. So you just need to check for the other cases.