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?

$$\lim\limits_{x\to \infty} \left(\frac{\ln (2x)}{\ln x}\right)^{\ln x} $$
$$=\lim\limits_{x\to \infty} \left(\frac{\ln 2+\ln x}{\ln x}\right)^{\ln x} $$
$$=\lim\limits_{x\to \infty} \left(\frac{\ln 2}{\ln x}+1\right)^{\ln x} $$
$$=\lim\limits_{x\to \infty} \left(\frac{1}{\frac{\ln x}{\ln 2}}+1\right)^{\ln x} $$
Let $n=\frac{\ln x}{\ln 2}$, then
$$\lim\limits_{n\to \infty} \left(1+\frac{1}{n}\right)^{n\ln 2} $$
$$=\left(\lim\limits_{n\to \infty} \left(1+\frac{1}{n}\right)^n\right)^{\ln 2} $$
$$=e^{\ln 2}=2$$
