semi-log plot

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

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

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

:$\backslash log\_\{n\}(F(x))\; =\; mx\; +\; b$

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

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:

: $m\; =\; \backslash frac\; \{F\_1\; -\; F\_0\}\{\backslash log\_n\; (x\_1\; /\; x\_0)\}$

which leads to

: $F\_1\; -\; F\_0\; =\; m\; \backslash log\_n\; (x\_1\; /\; x\_0)$

or

: $F\_1\; =\; m\; \backslash log\_n\; (x\_1\; /\; x\_0)\; +\; F\_0\; =\; m\; \backslash log\_n\; (x\_1)\; -\; m\; \backslash log\_n\; (x\_0)\; +\; F\_0$

which means that

$$F(x)\; =\; m\; \backslash log\_n\; (x)\; +\; \backslash mathrm\{constant\}$$

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:

: $F(x)\; =\; (F\_1\; -\; F\_0)\; \{\backslash left[\backslash frac\{\backslash log\_n\; (x\; /\; x\_0)\}\{\backslash log\_n(x\_1\; /\; x\_0)\}\backslash right]\}\; +\; F\_0\; =\; (F\_1\; -\; F\_0)\; \backslash log\_\{\backslash frac\{x\_1\}\{x\_0\}\}\{\backslash left(\backslash frac\{x\}\{x\_0\}\backslash right)\}\; +\; F\_0$

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:

: $m\; =\; \backslash frac\; \{\backslash log\_n\; (F\_1\; /\; F\_0)\}\{x\_1\; -\; x\_0\}$

which leads to

: $\backslash log\_n(F\_1\; /\; F\_0)\; =\; m\; (x\_1\; -\; x\_0)$

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

: $\backslash frac\{F\_1\}\{F\_0\}\; =\; n^\{m(x\_1\; -\; x\_0)\}$

or

: $F\_1\; =\; F\_0n^\{m(x\_1\; -\; x\_0)\}$

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

: $F(x)\; =\; \{F\_0\}\; n^\{\backslash left(\backslash frac\; \{x-x\_0\}\{x\_1-x\_0\}\backslash right)\; \backslash log\_n\; (F\_1\; /\; F\_0)\}$

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:

File:Phase diagram of water.svg of water. The Roman numerals indicate various ice phases.]]

While ten is the most common base, there are times when other bases are more appropriate, as in this example:{{Explain|date=April 2023|reason=No reason given for why base 2 is more appropriate in this case.}}

File:Influenza-2009-cases-logarithmic.png.]]

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.

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.

File:Bacterial growth en.svg

• 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

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