Need to prove that the equation has only 1 solution. I have been trying to solve the following equation:
$5^x+7^x=12^x.$
Obviously, x=1 is a solution but how do I prove that there are no other solutions.

we'll show it has no solutions for $x > 1$, hope you can use the idea to deal with the other case. your observation (that $1$ is a solution) will be crucial. write $x = 1 + y$ with $y > 0$, then
$$ 5^x + 7^x = 5 \cdot 5^y + 7 \cdot 7^y < (5+7)\cdot 7^y < 12 \cdot 12^y = 12^x$$
inequalities are strict because $y > 0$.
