Inscience andengineering, asemi-log plot/graph orsemi-logarithmicplot/graph has one axis on alogarithmic scale, the other on alinear scale. It is useful for data withexponential relationships, where onevariable covers a large range of values.^[1]

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

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

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

${\displaystyle \log (y)=(\gamma \log (a))x+\log (\lambda ).}$

Alog–linear (sometimes log–lin) plot has the logarithmic scale on they-axis, and alinear scale on thex-axis; alinear–log (sometimes lin–log) is the opposite. The naming isoutput–input (y–x), the opposite order from (x,y).

On a semi-log plot the spacing of the scale on they-axis (orx-axis) is proportional to the logarithm of the number, not the number itself. It is equivalent to converting they values (orx values) to their log, and plotting the data on linear scales. Alog–log plot uses the logarithmic scale for both axes, and hence is not a semi-log plot.

The equation of a line on a linear–log plot, where theabscissa axis is scaled logarithmically (with a logarithmic base ofn), would be

${\displaystyle F(x)=m{\log }_n(x)+b.}$

The equation for a line on a log–linear plot, with anordinate axis logarithmically scaled (with a logarithmic base ofn), would be:

${\displaystyle {\log }_n(F(x))=mx+b}$

${\displaystyle F(x)=n^{mx+b}=(n^{mx})(n^b).}$

On a linear–log plot, pick somefixed point (x₀,F₀), whereF₀ is shorthand forF(x₀), somewhere on the straight line in the above graph, and further some otherarbitrary point (x₁,F₁) on the same graph. The slope formula of the plot is:

${\displaystyle m=\frac{F_1-F_0}{{\log }_n(x_1/x_0)}}$

which leads to

${\displaystyle F_1-F_0=m{\log }_n(x_1/x_0)}$

or

${\displaystyle F_1=m{\log }_n(x_1/x_0)+F_0=m{\log }_n(x_1)-m{\log }_n(x_0)+F_0}$

which means that$${\displaystyle F(x)=m{\log }_n(x)+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}}$$

In other words,F is proportional to the logarithm ofx 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:

${\displaystyle F(x)=(F_1-F_0)\left[\frac{{\log }_n(x/x_0)}{{\log }_n(x_1/x_0)}\right]+F_0=(F_1-F_0){\log }_{\frac{x_1}{x_0}}\left(\frac{x}{x_0}\right)+F_0}$

On a log–linear plot (logarithmic scale on the y-axis), pick somefixed point (x₀,F₀), whereF₀ is shorthand forF(x₀), somewhere on the straight line in the above graph, and further some otherarbitrary point (x₁,F₁) on the same graph. The slope formula of the plot is:

${\displaystyle m=\frac{{\log }_n(F_1/F_0)}{x_1-x_0}}$

which leads to

${\displaystyle {\log }_n(F_1/F_0)=m(x_1-x_0)}$

Notice thatn^log_n(F₁) =F₁. Therefore, the logs can be inverted to find:

${\displaystyle \frac{F_1}{F_0}=n^{m(x_1-x_0)}}$

or

${\displaystyle F_1=F_0{n}^{m(x_1-x_0)}}$

This can be generalized for any point, instead of justF₁:

${\displaystyle F(x)=F_0{n}^{\left(\frac{x-x_0}{x_1-x_0}\right){\log }_n(F_1/F_0)}}$

Inphysics andchemistry, a plot of logarithm of pressure against temperature can be used to illustrate the variousphases of a substance, as in the following forwater:

While ten is the most commonbase, there are times when other bases are more appropriate, as in this example:^{[further explanation needed]}

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.

Inbiology andbiological engineering, the change in numbers ofmicrobes due toasexual 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 ofbacteria or other microbe as the dependent variable. This forms a plot with four distinct phases, as shown below.

• Nomogram, a diagram designed to approximate a function graphically
• Nonlinear regression#Transformation, for converting a nonlinear form to a semi-log form amenable to non-iterative calculation
• Log–log plot

• Charts
• Technical drawing
• Statistical charts and diagrams

• Articles with short description
• Short description is different from Wikidata
• Wikipedia articles needing clarification from April 2023