Is this a valid method of solving $3^x + 2^x = 35$? I want to know if this is a valid method of solving this equation:
$3^x + 2^x = 35$
$3^x+2^x = (7)(5)$
$3^x+2^x = (3+2^2)(2+3)$
$3^x+2^x = 3^2 + 2^3 + 18$
$3^x+2^x = 9 + 18 + 2^3 $
$3^x+2^x = 27 + 2^3 $
$3^x+2^x = 3^3 + 2^3 $
And now comes my problem. Is it correct to say that this last equation implies $x=3$? I don't think I can just compare terms just like it was a polynomial...
If this is not correct, is there a way of solving this equation algebraically?

The claim is equivalent to 
$$
f(x)=f(3)\implies x=3
$$
where $f(x)=3^x+2^x$. This is true because $f$ is injective (since it is strictly increasing for example).
