Find the unknown in the following problem. 
Given $$\lim_{x\to 0^+} \ln(x \ln(a))\ln\left(\frac{\ln(ax)}{\ln(x/a)}\right) = 6$$ find the value of $a$. 

My approach - I figured out that I can't do much with the first term as I cant tend that to 0. I manipulated 2nd term of log by adding and subtracting 1 and then multiplying and dividing by the suitable term to get the general form of log(1+x)/x. I'm stuck afterwards. Any hint will be awesome.

Let $f(x) = \ln(a) - \ln(x)$.
$\textit{(Where does $f(x)$ go as $x \rightarrow 0^+$ ?)}$
Then
\begin{align} \ln(x \ln(a)) \ln\left[\frac{\ln(ax)}{\ln(\frac{x}{a})}\right] & = \left[\ln(a) + \ln(\ln(a)) - f(x) \right]\ln\left[1+\frac{-2\ln(a)}{f(x)} \right]\\\\ & = \left[\ln(a) + \ln(\ln(a))\right]\ln\left[1+\frac{-2\ln(a)}{f(x)} \right]-\ln\left[\left(1+\frac{-2\ln(a)}{f(x)}\right)^{f(x)} \right]  &\end{align}
Then, to take the limit would ...
