So I have a real system that for a given setting, x, returns a value, y. These values appear to follow (with some limits) the form of $y = ax^{-b}$ - could also be expressed as $y = \frac{a}{x^b} $.
How many data pairs do I need to be able to work out what $a$ and $b$ are, and how do I work them out? Excel seems to have no trouble, but I also have no idea how it gets there.
You will need more than two pairs to do a least square fit.
Transform the equation to $$\ln{y}=a-b\ln{x}$$
Then you can use usual least square method to find the parameter $a,-b$, using the pairs $(\ln{x_i},\ln{y_i})$.