## Sunday, 14 April 2013

### Logarithmic Graphs

What is it? A graph with a logarithmic scale, one which increases by multiplications in value rather than additions (e.g. 1, 10, 100, 1000 rather than 1, 2, 3, 4). The value by which the scale is multiplied by is usually 10 (i.e. log base 10). Both scales may be logarithmic or just one (semi-logarithmic graph). Semi-logarithmic graphs normally have time displayed on x

Benefits? Useful for studying data that changes exponentially. Can display a much larger range of data than an arithmetic scale. Allows one to see increased detail at smaller values, while larger values are compressed. Small values occupy a larger proportion of the scale in comparison with larger values. On an arithmetic scale, unless the graph paper was very large, the smaller values would appear too small to see properly. Allows comparison between trends in small and large values

Useful for showing rate of change. A steep gradient shows a fast rate of change while a shallow gradient represents a slowing rate of change.

A constant proportional rate of change (an exponential change) is represented by a straight line on a logarithmic graph (rather than a curved line on a arithmetic graph). This means logarithmic graphs are good for comparing rates of change.

Limitations? Zero cannot be plotted. Positive and negative values cannot be plotted on the same graph. Can be difficult to draw and interpret as scale is distorted.

Uses in geography?
Studying population data e.g. as in Gapminder
The Hjulstrom curve.
The Richter magnitude scale/The Moment magnitude scale
Magnitude-frequency flood risk analysis.

Example 1 (Logarithmic)

Example 2 (Semi-logarithmic)
nb - in this graph the scale is multiplied by 2 each time rather than 10 (i.e. log base 2).

Further information
Check out this video for a great explanation of logarithmic scales!
You can also find an introduction to logarithms here