Rearranging into $y=mx+c$ format and finding unknowns $a$ and $b$ 
Two quantities $x$ and $y$ are connected by a law $y = \frac{a}{1-bx^2}$ where $a$ and $b$ are constants. Experimental values of $x$ and $y$ are given in the table below:
  $$
\begin{array}{|l|l|l|l|l|l|}
\hline
x & 6 &8 & 10 & 11 & 12\\
\hline
y & 5.50 & 6.76 & 9.10 & 11.60 & 16.67\\\hline 
\end{array}
$$
  By plotting a suitable graph, find $a$ and $b$. (Use tables correct to $2$ significant figures in your work)

I dont know whether or not i should use logs to rearrange the equation into $y=mx+c$ format. 

You don't need to rearrange your data. If you know that the physical law is $y=\frac{a}{1-bx^2}$, you can just find the parameters $a,b$ such that the squared error:
$$ f(a,b)=\sum_{i=1}^{n}\left(y_i-\frac{a}{1-bx_i^2}\right)^2 $$
attains its minimum by annihilating $\frac{\partial f}{\partial a}$ and $\frac{\partial f}{\partial b}$ like we usually do when dealing with linear regressions. In this case, however, computations can be messier. As pointed by Claude Leibovici in the comments, the only problem is that you face a nonlinear regression and then you need reasonable estimates to start. What basically they ask is to find a way to get good estimates by some transform of the data. By doing what he suggested (solving a linear regression with respect to the variables $\frac{1}{y},x^2$), the first step lead to $a=4.52218, b=0.0050554$ from which the nonlinear regression can start and converge very fast to $a=4.50954, b=0.0050637$.
