Log Graphs

There are times when we don't just want to plot one set of data against another. Instead, we realise that there exists a certain relationship between the variables, and want to know what it is. For example, let us deal with the length, l of an oscillating beam, and its period, T of oscillation. Assume that they are related to each other as follows: T = klⁿ, where k and n are unknown constants, not necessarily integers.

How do we go about finding what k and n are using a graph?

I am assuming that you have a basic knowledge of what logs are, and don't consider them to grow on trees. Fullick (p. 39) has a section if you're not au fait.

Take logs of both sides of the equation:

log T = log (klⁿ)

log T = log k + log lⁿ

log T = log k + n log l

Compare this to the standard way of looking at an x-y graph, i.e. y = mx + c, and we find the following direct comparisons:

y = log T, x = log l

m = n, c = log k

Hence, a graph of logT against logl will yield a straight-line relationship, gradient n and y-intercept logk.

Thus any physical quantities that appear to have a power relationship between them can have a formula derived directly from the logs of the data! The other advantage of plotting a log graph is that you are, within experimental uncertainty, guaranteed a straight line - hence it is an easy matter to plot it by hand if you do not have access to a computer to plot the graphs on.

For further information on better graph plotting, and help with coursework report writing in general, see the rest of the SciRep series.