Reference articles on history, science, culture and more
Surprise meSurprise meSubscribe
Encyclopedia

Semi-log plot

Type of graph

Wikipedia contributorsAdapted for Fathomly · View authors
3 min read
Source updated July 18, 2026
Image credit is listed at the end of this article.

In science and engineering, a semi-log plot/graph or semi-logarithmic plot/graph has one axis on a logarithmic scale, the other on a linear scale. It is useful for data with exponential relationships, where one variable covers a large range of values.

All equations of the form y=\lambda a^{\gamma x} form straight lines when plotted semi-logarithmically, since taking logs of both sides gives

\log _{a}y=\gamma x+\log _{a}\lambda .

This is a line with slope \gamma and \log _{a}\lambda vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2:

\log(y)=(\gamma \log(a))x+\log(\lambda ).

A log-linear (sometimes log-lin) plot has the logarithmic scale on the y-axis, and a linear scale on the x-axis; a linear-log (sometimes lin-log) is the opposite. 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 linear scales. A log-log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.

01Equations

The equation of a line on a linear-log plot, where the abscissa axis is scaled logarithmically (with a logarithmic base of n), would be

F(x)=m\log _{n}(x)+b.\,

The equation for a line on a log-linear plot, with an ordinate axis logarithmically scaled (with a logarithmic base of n), would be:

\log _{n}(F(x))=mx+b
F(x)=n^{mx+b}=(n^{mx})(n^{b}).

Finding the function from the semi-log plot

Linear-log plot

On a linear-log plot, 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:

m={\frac {F_{1}-F_{0}}{\log _{n}(x_{1}/x_{0})}}

which leads to

F_{1}-F_{0}=m\log _{n}(x_{1}/x_{0})

or

F_{1}=m\log _{n}(x_{1}/x_{0})+F_{0}=m\log _{n}(x_{1})-m\log _{n}(x_{0})+F_{0}

which means that F(x)=m\log _{n}(x)+\mathrm {constant}

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:

F(x)=(F_{1}-F_{0}){\left[{\frac {\log _{n}(x/x_{0})}{\log _{n}(x_{1}/x_{0})}}\right]}+F_{0}=(F_{1}-F_{0})\log _{\frac {x_{1}}{x_{0}}}{\left({\frac {x}{x_{0}}}\right)}+F_{0}

log-linear plot

On a log-linear 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:

m={\frac {\log _{n}(F_{1}/F_{0})}{x_{1}-x_{0}}}

which leads to

\log _{n}(F_{1}/F_{0})=m(x_{1}-x_{0})

Notice that nlogn(F1) = F1. Therefore, the logs can be inverted to find:

{\frac {F_{1}}{F_{0}}}=n^{m(x_{1}-x_{0})}

or

F_{1}=F_{0}n^{m(x_{1}-x_{0})}

This can be generalized for any point, instead of just F1:

F(x)={F_{0}}n^{\left({\frac {x-x_{0}}{x_{1}-x_{0}}}\right)\log _{n}(F_{1}/F_{0})}
The linear-log type of a semi-log graph, defined by a logarithmic scale on the x axis, and a linear scale on the y axis. Plotted lines are: y = 10x (red), y = x (green), y = log(x) (blue).
The linear-log type of a semi-log graph, defined by a logarithmic scale on the x axis, and a linear scale on the y axis. Plotted lines are: y = 10x (red), y = x (green), y = log(x) (blue).
log-linear pressure-temperature phase diagram of water. The Roman numerals indicate various ice phases.
log-linear pressure-temperature phase diagram of water. The Roman numerals indicate various ice phases.

02Real-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:

Notice that while the horizontal (time) axis is linear, with the dates evenly spaced, the vertical (cases) 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, i.e. 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, e.g. social distancing.

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.

A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1).
A semi-logarithmic plot of cases and deaths in the 2009 outbreak of influenza A (H1N1).
Bacterial growth curve
Bacterial growth curve
Watch videos about Semi-log plotExplainers and documentaries on YouTube (opens in a new tab)

Sources and credits

This article is adapted from the Wikipedia article “Semi-log plot”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

Images, from Wikimedia Commons:

Fathomly is not affiliated with or endorsed by the Wikimedia Foundation. Spotted a problem? Tell us.

Continue exploring

Related topics

Nomogram

A nomogram 'law' and γράμμα (gramma) 'that which is drawn'), also called a nomograph, alignment chart, or abac, is a graphical calculating device, a two-dimensional diagram designed to allow the approximate graphical computation of a mathematical function. The field of nomography was invented in 1884 by the French engineer Philbert Maurice d'Ocagne (1862-1938) and used extensively for many years to provide engineers with fast graphical calculations of complicated formulas to a practical precision.

Nonlinear regression

In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.

Log-log plot

In science and engineering, a log-log graph or log-log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions, relationships of the form y = a x k {\displaystyle y=ax^{k}}, appear as straight lines in a log-log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept.