Hello
let : $ X=2^x$ and $ Y=3^x$
and $ (E): 9^x+4^x+1=6^x+3^x+2^x$
$ (E) \Leftrightarrow X^2+Y^2 +1 = XY+X+Y$
on the other hand : $ X^2+Y^2 +1\geq 2XY+1$
Hence , $ (X-1)(1-Y) \geq 0$
therefore : $ X-1 \leq 0$ and $ Y-1 \geq 0$
Or $ X-1 \geq 0$and $ Y-1 \leq 0$
From That inequalities : $ x=0$ By introducing expononciel function ( $ 2^x= e^{xln(2)}$ )
*Excuse me for me poor english