Answer
The sequence converges to $1$.
Work Step by Step
We can write $\ln{2n}$ as $\ln{2} + \ln{n}$. Hence
$$a_n = \frac{\ln{n}}{\ln{2} + \ln{n}} = \frac{1}{\frac{\ln{2}}{\ln{n}} + 1}.$$
Since the fraction in the denominator approaches 0 as $n \to \infty$, our sequence converges to $\frac{1}{0 + 1} = 1$.
