Semi-log plot

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:

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

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.

Equations

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

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

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

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

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

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:

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

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 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:

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

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:

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

which leads to

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

Notice that n^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 just F₁:

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