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.^[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:

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

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

$${\displaystyle F(x)=m\log _{n}(x)+\mathrm {constant} }$$

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:

${\displaystyle F(x)=(F_{1}-F_{0}){\left}+F_{0}=(F_{1}-F_{0})\log _{\frac {x_{1}}{x_{0}}}{\left({\frac {x}{x_{0}}}\right)}+F_{0}}$

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

${\displaystyle \log _{n}(F_{1}/F_{0})=m(x_{1}-x_{0})}$Zdroj:https://en.wikipedia.org?pojem=Log-lin_plot
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