Interesting limit involving hordes of logarithms. 
$$\lim_{x\to 0^{+}}\ln (x\ln a)\ln \left(\dfrac{\ln ax}{\ln\frac xa}\right)=6$$
  Find the value of $a$.
Answer: $e^3$

This struck me as an interesting problem and wanted to know if there are any more methods to solve it except the one I have used (written as an answer below).
Thanks!
Edit: for those requiring additional context, I have SOLVED the problem and my solution is posted below as an answer. I am merely looking for other ways to solve this question. 

$$\frac{\log ax}{\log\dfrac xa}=1+\frac{2\log a}{\log\dfrac xa}$$ so that
$$\log\frac{\log ax}{\log\dfrac xa}\sim\frac{2\log a}{\log\dfrac xa}.$$
As the denominator will simplify with $\log(x\log a))$, we are left with 
$$2\log a=6.$$
