Limit to infinity with natural logarithms $\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $ I found the following problem in my calculus book:
Solve:  
$$\lim_{x\to \infty } \left(\frac{\ln (2 x)}{\ln (x)}\right)^{\ln (x)} $$
I tried to solve it using log rules and l'Hôpital's rule with no success, can someone give me any hints on how to go about this?

HINT: Using Sum of Logarithms,
$$\dfrac{\ln2x}{\ln x}=1+\dfrac{\ln2}{\ln x}$$
and set $\dfrac{\ln2}{\ln x}=n$ in $$\lim_{n\to\infty}\left(1+\dfrac1n\right)^n=e$$
Finally if $a^x=M$ from the definition, $x=\log_aM$
$$\displaystyle\implies a^{\log_aM}=M$$ (See also)
