Semi-log graph

The log-lin type of a semi-log graph, defined by a logarithmic scale on the y axis, and a linear scale on the x axis. Plotted lines are: y=x (green), y=10^x(red), y=log(x) (blue).

The lin-log type of a semi-log graph, defined by a logarithmic scale on the x axis, and a linear scale on the y axis. Plotted lines are: y=x (green), y=10^x(red), y=log(x) (blue).

In science and engineering, a semi-log graph or semi-log plot is a way of visualizing data that are changing with an exponential relationship. One axis is plotted on a logarithmic scale. This kind of plot is useful when one of the variables being plotted covers a large range of values and the other has only a restricted range – the advantage being that it can bring out features in the data that would not easily be seen if both variables had been plotted linearly. ^[1]

All functions of the form y = λa^γx form straight lines, since taking logs of both sides is equal to

log_ay = γx + log_aλ.

This can easily be seen as a line in slope-intercept form with γ as slope, log_aλ as the y-intercept. To facilitate use with logarithmic tables, one usually takes logs to base 10 or e:

log(y) = (γlog(a))x + log(λ).

The term log-lin is used to describe a semi-log plot with a logarithmic scale on the y-axis, and a linear scale on the x-axis. Likewise, a lin-log graph uses a logarithmic scale on the x-axis, and a linear scale on the y-axis.

A log-log graph uses the logarithmic scale for both axes, and hence is not a semi-log graph.

on semi-log graph the spacing of the scale on the y-axis is proportional to the logarithm of the number, not the number itself. it is the equvalent of converting the Y values to their log, and plotting the data on linear graph paper.

Real-world example: 2009 "swine flu" progression

While ten is the most common base, there are times when other bases are more appropriate, as in this example:

A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1). Notice that while the horizontal (time) axis is linear, with the dates evenly spaced, the vertical (cases) axis is logarithmic, with the evenly spaced divisions being labelled with successive powers of two.

See also

• Nomograph, more complicated graphs

References

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