Solve equations like $3^x+4^x=7^x$ How can I solve something like this?
$$3^x+4^x=7^x$$
I know that $x=1$, but I don't know how to find it. Thank you!

Defining $f(x):=\mathrm{log}_7\left(3^x+4^x\right)$, you want to search for fixed points of $f$. But
$$
f^\prime(x)=\frac{1}{\ln 7}\cdot \frac{3^x\ln 3+4^x \ln 4}{3^x+4^x}>0
$$
and
$$
f^{\prime\prime}(x)=\frac{1}{\ln 7}\cdot \frac{3^x4^x}{(3^x+4^x)^2} \cdot ((\ln 4)^2+(\ln 3)^2-2\ln 3 \ln 4)
$$
which is positive too by arithmetic-geometric mean inequality.
