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 for a line with an ordinate axis logarithmically scaled would be:
The equation of a line on a plot where the abscissa axis is scaled logarithmically would be
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.
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