Log–log plot
plot that uses logarithmic scales on both the horizontal and vertical axes
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).
This brief starts where responsible research should: with the source description of “Log–log plot” as plot that uses logarithmic scales on both the horizontal and vertical axes. Everything that follows is an evidence route, not borrowed authority.
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