I am really stumped on this question. I have tried re arranging it but I can't seem to get it right.
ln x - ln x^2 = ln 27
At what point do you get stuck? Are you able to combine the left-hand side into a single logarithm? After that you can equate the expressions inside the logs (or, equivalently, make both sides exponential to the base $\displaystyle e$ so that the logarithm operations get canceled out).
With the rule $\displaystyle \ln(a)-\ln(b)=\ln\left(\frac{a}{b}\right)$ the equation is equivalent to:
$\displaystyle \ln\left(\frac{x}{x^2}\right)=\ln(27)$
$\displaystyle \Leftrightarrow \ln\left(\frac{1}{x}\right)=\ln(27)$
Therefore $\displaystyle \frac{1}{x}=27$ and thus $\displaystyle x=\frac{1}{27}$ is the solution.