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Type of graph

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]} or to zoom in and visualize that - what seems to be a straight line in the beginning - is in fact the slow start of a logarithmic curve that is about to spike and changes are much bigger than thought initially.^{[2]}

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.

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:

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

On a linear-log plot, pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) 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

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

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 (*F*_{0}, *x*_{0}) and (*F*_{1}, *x*_{1}) will have the function:

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

On a log–linear plot (logarithmic scale on the y-axis), pick some *fixed point* (*x*_{0}, *F*_{0}), where *F*_{0} is shorthand for *F*(*x*_{0}), somewhere on the straight line in the above graph, and further some other *arbitrary point* (*x*_{1}, *F*_{1}) 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 *n*^{logn(F1)} = *F*_{1}. 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 *F _{1}*:

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

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:

While ten is the most common base, there are times when other bases are more appropriate, as in this example:

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.

- 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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