A linear relationship is the easiest relationship to decode ie. find a relationship for. It may be that even if two quantities are not in a linear relationship, functions of the quantities can be found that do bear a linear relationship. We may then plot graphs of the functions, and if the relationship appears linear we may write down a linear relationship between the two functions over the range of observations.

In attempting to find a straight line relationship we may try logarithmic, reciprocal or power relationships. We can plot products of powers ofandagainst logs of functions ofand We are looking to obtain a straight line relationship. Towards this end we make the following observations:

If the graph curves up the takingor a root of– or equivalently, exponentiatingor raisingto some power – will tend to straighten the curve out, but often some foresight is needed. There is a positive relationship betweenandin the table below,

since both increase together, but the relationship is not linear since the gradient between

is not the same as for

We can take the reciprocals of bothandand plotagainstThe transformed data is

The graph below shows an excellent fit to a straight line with equation