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. All equations of the form y = λ a γ x {\displaystyle y=\lambda a^{\gamma x}} form straight lines when plotted semi-logarithmically, since taking logs of both sides gives
log a y = γ x + log a λ . {\displaystyle \log _{a}y=\gamma x+\log _{a}\lambda .}
This is a line with slope γ {\displaystyle \gamma } and log a λ {\displaystyle \log _{a}\lambda } vertical intercept. The logarithmic scale is usually labeled in base 10; occasionally in base 2:
log ( y ) = ( γ log ( a ) ) x + log ( λ ) . {\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
F ( x ) = m log n ( x ) + b . {\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:
log n ( F ( x ) ) = m x + b {\displaystyle \log _{n}(F(x))=mx+b}
F ( x ) = n m x + b = ( n m x ) ( n 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:
m = F 1 − F 0 log n ( x 1 / x 0 ) {\displaystyle 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 ) {\displaystyle 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 {\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
F ( x ) = m log n ( x ) + c o n s t a n t {\displaystyle F(x)=m\log _{n}(x)+\mathrm {constant} }
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