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The function $f(x)=4^x+6^x-9^x$ is such that $f(0)=1>0, f(1)=1>0, f(2)=-29$ and next $g(x)=(4/9)^x+(6/9)^x-1 \implies f'(x)<0$ for all real values of $x$. So $g(x)$ being monotonic the equation $$4^x+6^x=9^x$$ has exactly one real solution. The question is whether this real root can be found analytically by hand.