If
\[
\int_{x}^{2x}\left(1+(\log 2)\log x\right)^{x}\,dx=f(x)+c
\]
and
\[
f(1)=0,
\]
then \(f(e)=\)
Show Hint
Whenever a definite or indefinite integral defines a function,
\[
f(x)=\int g(x)\,dx,
\]
first simplify the integrand as much as possible and then use the given initial condition to determine the constant of integration.