This page uses content from Wikipedia and is licensed under CC BY-SA.

In science and engineering, a **semi-log graph** or **semi-log plot** is a way of visualizing data that are related according to an exponential relationship. One axis is plotted on a logarithmic scale.

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.^{[1]}

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:

On a lin-log plot (logarithmic scale on the x-axis), pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) 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 (*F*_{0}, *x*_{0}) and (*F*_{1}, *x*_{1}) will have the function:

On a log-lin plot (logarithmic scale on the y-axis), pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) on the same graph. The slope formula of the plot is:

which leads to

Notice that *n*^{logn(F1)} = *F*_{1}. Therefore, the logs can be inverted to find:

or

This can be generalized for any point, instead of just *F _{1}*:

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:

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