In science and engineering, a semi-log plot, or semi-log graph, has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, or where one variable covers a large range of values. All equations of the form form straight lines when plotted semi-logarithmically, since taking logs of both sides gives This is a line with slope and vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2: A log-linear plot has the logarithmic scale on the y-axis, and a linear scale on the x-axis; a linear-log is the opposite. The naming is output-input, the opposite order from. On a semi-log plot the spacing of the scale on the y-axis is proportional to the logarithm of the number, not the number itself. It is equivalent to converting the y values to their log, and plotting the data on linear scales. A log-log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.
Equations
The equation of a line on a linear-log plot, where the abscissa axis is scaled logarithmically, would be The equation for a line on a log-linear plot, with an ordinate axis logarithmically scaled, would be:
Finding the function from the semi–log plot
Linear-log plot
On a linear-log plot, pick some fixed point, where F0 is shorthand for F, somewhere on the straight line in the above graph, and further some other arbitrary point on the same graph. The slope formula of the plot is: which leads to or 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 and will have the function:
log-linear plot
On a log-linear plot, pick some fixed point, where F0 is shorthand for F, somewhere on the straight line in the above graph, and further some other arbitrary point on the same graph. The slope formula of the plot is: which leads to Notice that nlogn = F1. Therefore, the logs can be inverted to find: or This can be generalized for any point, instead of just F1:
In physics and chemistry, a plot of logarithm of pressure against temperature can be used to illustrate the various phases of a substance, as in the following for water: of water. The Roman numerals indicate various ice phases.
2009 "swine flu" progression
While ten is the most common base, there are times when other bases are more appropriate, as in this example: . Notice that while the horizontal axis is linear, with the dates evenly spaced, the vertical axis is logarithmic, with the evenly spaced divisions being labelled with successive powers of two. The semi-log plot makes it easier to see when the infection has stopped spreading at its maximum rate ie. the straight line on this exponential plot, and starts to curve to indicate a slower rate. This might indicate that some form of mitigation action is working eg. social distancing.
