In science and engineering, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses logarithmic scales on both the horizontal and vertical axes. Power functions – relationships of the form y = a x k {\displaystyle y=ax^{k}} – appear as straight lines in a log–log graph, with the exponent corresponding to the slope, and the coefficient corresponding to the intercept. Thus these graphs are very useful for recognizing these relationships and estimating parameters. Any base can be used for the logarithm, though most commonly base 10 (common logs) are used.
Relation with monomials Given a monomial equation y = a x k , {\displaystyle y=ax^{k},} taking the logarithm of the equation (with any base) yields:
log y = k log x + log a . {\displaystyle \log y=k\log x+\log a.}
Setting X = log x {\displaystyle X=\log x} and Y = log y , {\displaystyle Y=\log y,} which corresponds to using a log–log graph, yields the equation
Y = m X + b {\displaystyle Y=mX+b}
where m = k is the slope of the line (gradient) and b = log a is the intercept on the (log y)-axis, meaning where log x = 0, so, reversing the logs, a is the y value corresponding to x = 1.
Equations The equation for a line on a log–log scale would be:
log 10 F ( x ) = m log 10 x + b , {\displaystyle \log _{10}F(x)=m\log _{10}x+b,}
F ( x ) = x m ⋅ 10 b , {\displaystyle F(x)=x^{m}\cdot 10^{b},}
where m is the slope and b is the intercept point on the log plot.
Slope of a log–log plot
To find the slope of the plot, two points are selected on the x-axis, say x1 and x2. Using the below equation:
log [ F ( x 1 ) ] = m log ( x 1 ) + b , {\displaystyle \log[F(x_{1})]=m\log(x_{1})+b,}
and
log [ F ( x 2 ) ] = m log ( x 2 ) + b . {\displaystyle \log[F(x_{2})]=m\log(x_{2})+b.}
The slope m is found taking the difference:
m = log ( F 2 ) − log ( F 1 ) log ( x 2 ) − log ( x 1 ) = log ( F 2 / F 1 ) log ( x 2 / x 1 ) , {\displaystyle m={\frac {\log(F_{2})-\log(F_{1})}{\log(x_{2})-\log(x_{1})}}={\frac {\log(F_{2}/F_{1})}{\log(x_{2}/x_{1})}},}
where F1 is shorthand for F(x1) and F2 is shorthand for F(x2). The figure at right illustrates the formula. Notice that the slope in the example of the figure is negative. The formula also provides a negative slope, as can be seen from the following property of the logarithm:
log ( x 1 / x 2 ) = − log ( x 2 / x 1 ) . {\displaystyle \log(x_{1}/x_{2})=-\log(x_{2}/x_{1}).}
Finding the function from the log–log plot The above procedure now is reversed to find the form of the function F(x) using its (assumed) known log–log plot. To find the function F, 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. Then from the slope formula above:
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