I am currently stuck after this. I need to find a non-linear relationship to predict how log (X Variable) will affect log (Y Variable). Would simply using this log-log model be enough to predict? Or would I need to manipulate the model further by adding another term such as a squared log(x variable) term, much like in a quadratic model?
I think you should try to fit a polynomial regression because your data looks like a shaped distribution. Once you fit polynomial, you may want to compare it with the linear regression by likelihood-ratio test to convince yourself that your polynomial fitting is worth the loss of one or more degrees of freedom. Your polynomial model will always fit no worse than a simple linear model, the likelihood-ratio test will tell you whether it'd be a good idea to do so.
From the plot, it looks like a second-order polynomial should be sufficient.