# Semi-log plot

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 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 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.

## Equations

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

## Real-world examples

### Phase diagram of water

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:

### 2009 "swine flu" progression

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

### Microbial growth

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.

## See also

- Nomograph, more complicated graphs
- Nonlinear regression#Transformation, for converting a nonlinear form to a semi-log form amenable to non-iterative calculation