Exponential patterns on linear and log scales

What the exam asks

Expect to be handed a graph and asked to classify the pattern or read a rate off it. The commonest items are: decide from a table whether growth is linear or exponential, say what a straight line on a given pair of axes implies, and read a doubling time or a half-life off a semi-log plot. The trap that catches most candidates is treating a log axis as if it were linear: they see a point halfway between the 10 and the 100 gridline and call it 55. It is about 32, because the axis measures ratio. Check the axis labels before you read a single value; if consecutive labels are 1, 10, 100, 1000, every instinct you have about distance on that axis is wrong until you convert it to a factor.

An exponential pattern is one where a fixed time interval multiplies the quantity by a fixed factor. That is the whole definition, and everything else follows from it. Linear growth adds the same amount each step; exponential growth multiplies by the same amount each step. Bacterial counts, radioactive decay, drug clearance, compound interest and reaction rates against temperature are all posed this way.

A log axis exists because logs turn multiplication into addition. If a quantity multiplies by the same factor every interval, then its log gains the same amount every interval, and a constant gain per interval is a straight line. So an exponential plotted with a log vertical axis is straight, and the steepness of that line is the rate. This is why growth data is so often shown semi-log: it converts a shape you cannot read into one you can.

What you carry in is how to read the axis, not how to derive it. On a log axis equal spacing means equal ratio, not equal difference. One fixed distance up the page is always one factor of ten, whether you are travelling from 1 to 10 or from 10,000 to 100,000. Once you believe that, most of these questions are a two-point reading.

What to hold

• Exponential means a fixed interval multiplies the quantity by a fixed factor, so equal steps give equal ratios rather than equal differences.
• On a linear axis an exponential is a curve that steepens without bound, which makes early values unreadable and late values off the page.
• A log axis is spaced by ratio: equal distances represent equal multiplying factors, so each decade occupies the same height.
• An exponential is a straight line on a semi-log plot, which has a log vertical axis and a linear horizontal axis.
• A power law such as y proportional to x cubed is a straight line on a log-log plot, where both axes are log, and its slope is the power.
• The slope of a semi-log line is the rate: rising one decade per fixed interval means multiplying by ten in that interval.
• The midpoint between two values on a log axis is their geometric mean, so halfway between 10 and 1000 sits at 100, not at 505.
• Ten doublings give a factor of 1024, close enough to 1000, so roughly 3.3 doublings make up one decade.
• After n half-lives the fraction remaining is one half raised to the power n, and this is exponential decay: a straight line sloping down on a semi-log plot.
• A steepening exponential curve does not mean the growth rate is increasing: the fractional rate is constant, and it is only the absolute increment that grows.
• Gridlines within a decade of a log axis are bunched towards the top, because the gap from 1 to 2 is a bigger ratio than the gap from 8 to 9.
• pH, decibels and earthquake magnitude are log scales, so a one-unit change on any of them is a fixed multiplying factor rather than a fixed amount.