This kind of plotting method 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.
All equations of the form form straight lines when plotted semi-logarithmically, since taking logs of both sides gives
This can easily be seen as a line in slope-intercept form with as the slope and as the vertical intercept. To facilitate use with logarithmic tables, one usually takes logs to base 10 or e, or sometimes base 2:
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 plot uses a logarithmic scale on the x-axis, and a linear scale on the y-axis. Note that the naming is output-input (y-x), the opposite order from (x, y).
On a semi-log plot the spacing of the scale on the y-axis (or x-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values (or x values) to their log, and plotting the data on lin-lin scales. A log-log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.
The equation of a line on a lin-log plot, where the abscissa axis is scaled logarithmically (with a logarithmic base of n), would be
The equation for a line on a log-lin plot, with an ordinate axis logarithmically scaled (with a logarithmic base of n), would be:
Finding the function from the semi–log plot
On a lin-log plot (logarithmic scale on the x-axis), pick some fixed point (x0, F0), where F0 is shorthand for F(x0), somewhere on the straight line in the above graph, and further some other arbitrary point (x1, F1) on the same graph. The slope formula of the plot is:
which leads to
which means that
In other words, F is proportional to the logarithm of x times the slope of the straight line of its lin–log graph, plus a constant. Specifically, a straight line on a lin–log plot containing points (F0, x0) and (F1, x1) will have the function:
On a log-lin plot (logarithmic scale on the y-axis), pick some fixed point (x0, F0), where F0 is shorthand for F(x0), somewhere on the straight line in the above graph, and further some other arbitrary point (x1, F1) on the same graph. The slope formula of the plot is:
which leads to
Notice that nlogn(F1) = F1. Therefore, the logs can be inverted to find:
This can be generalized for any point, instead of just F1:
Phase diagram of water
2009 "swine flu" progression
While ten is the most common base, there are times when other bases are more appropriate, as in this example:
In biology and biological engineering, the change in numbers of microbes due to asexual reproduction and nutrient exhaustion is commonly illustrated by a semi-log plot. Time is usually the independent axis, with the logarithm of the number or mass of bacteria or other microbe as the dependent variable. This forms a plot with four distinct phases, as shown below.
- Multiplicative calculus
- Nomograph, more complicated graphs
- Nonlinear regression#Transformation, for converting a nonlinear form to a semi-log form amenable to non-iterative calculation
- Log–log plot