Lets say a scientist is studying the growth of a colony of bacteria. She has a hunch that the number of bacteria is given by the following:

n = 10^{kt},

where n is the number of bacteria in the colony, t is the time in minutes and k is a constant.

She does an experiment to measure the number of bacteria on a petri dish over time. At the start of the experiment there is 1 bacteria on the petri dish. The number of bacteria is measured every 10 minutes and a graph is plotted of the number of bacteria against time in minutes:

After 2 hours there are 4156 bacteria, which when plotted on the graph, makes the earlier numbers look close to zero. Is the postulated relationship between the number of bacteria and number of minutes the bacterial have been on the petri dish correct? How can the scientist find the value of the constant k? This is a very difficult question to answer using this plot.

Lets go back to the equation n = 10^{kt}. It can be re-written as:

log_{10}n = kt

So, if a graph of log_{10}n against t is plotted, it should be a straight line passing through the origin with a gradient of k.

Here is a plot of ten times the logarithm of the number of bacteria against time:

There is a clear linear (straight line) relationship between the x and y, give or take some experimental error. If a straight line is fitted to the data, it might look like this:

The gradient is 0.295, so the constant k is 0.0295 and the relationship between the number of bacteria and time is:

n = 10^{0.0295t},

which has been plotted in purple along with the experimental data in red:

This is almost like waving a magic wand over the problem! The use of logarithms has transformed this problem from something virtually intractable to the relatively simple problem of analysing whether the data is a good fit to a straight line and if it is, then measuring the gradient of that line in order to obtain the value for k.

This is obviously a contrived problem with made-up data, but bacteria populations do grow exponentially and there are plenty of other real world examples where the relationship between two quantities involves multiple orders of magnitude. Examples are sound measurements, growth of debts/savings subject to interest rates and earthquake intensities.