In
science
Science is a systematic endeavor that builds and organizes knowledge in the form of testable explanations and predictions about the universe.
Science may be as old as the human species, and some of the earliest archeological evidence for ...
and
engineering
Engineering is the use of scientific method, scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...
, a log–log graph or log–log plot is a two-dimensional graph of numerical data that uses
logarithmic scale
A logarithmic scale (or log scale) is a way of displaying numerical data over a very wide range of values in a compact way—typically the largest numbers in the data are hundreds or even thousands of times larger than the smallest numbers. Such a ...
s on both the horizontal and vertical axes.
Power functions – relationships of the form
– 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
taking the logarithm of the equation (with any base) yields:
Setting
and
which corresponds to using a log–log graph, yields the equation:
where ''m'' = ''k'' is the slope of the line (
gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
) 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.
M. Bourne ''Graphs on Logarithmic and Semi-Logarithmic Paper'' (www.intmath.com)
/ref>
Equations
The equation for a line on a log–log scale would be:
:
:
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 ''x''1 and ''x''2. Using the above equation:
:
and
:
The slope ''m'' is found taking the difference:
:
where ''F''1 is shorthand for ''F''(''x''1) and ''F''2 is shorthand for ''F''(''x''2). 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:
:
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'' (''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. Then from the slope formula above:
:
which leads to
:
Notice that 10log10(''F''1) = ''F''1. Therefore, the logs can be inverted to find:
:
or
:
which means that
:
In other words, ''F'' is proportional to ''x'' to the power of the slope of the straight line of its log–log graph. Specifically, a straight line on a log–log plot containing points (''F''0, ''x''0) and (''F''1, ''x''1) will have the function:
:
Of course, the inverse is true too: any function of the form
:
will have a straight line as its log–log graph representation, where the slope of the line is ''m''.
Finding the area under a straight-line segment of log–log plot
To calculate the area under a continuous, straight-line segment of a log–log plot (or estimating an area of an almost-straight line), take the function defined previously
and integrate it. Since it is only operating on a definite integral (two defined endpoints), the area A under the plot takes the form
Rearranging the original equation and plugging in the fixed point values, it is found that
Substituting back into the integral, you find that for A over ''x''0 to ''x''1
Therefore:
For ''m'' = −1, the integral becomes
Applications
These graphs are useful when the parameters ''a'' and ''b'' need to be estimated from numerical data. Specifications such as this are used frequently in economics
Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and intera ...
.
One example is the estimation of money demand
In monetary economics, the demand for money is the desired holding of financial assets in the form of money: that is, cash or bank deposits rather than investments. It can refer to the demand for money narrowly defined as M1 (directly spendable ...
functions based on inventory theory Material theory (or more formally the mathematical theory of inventory and production) is the sub-specialty within operations research and operations management that is concerned with the design of production/ inventory systems to minimize costs: i ...
, in which it can be assumed that money demand at time ''t'' is given by
:
where ''M'' is the real quantity of money
Money is any item or verifiable record that is generally accepted as payment for goods and services and repayment of debts, such as taxes, in a particular country or socio-economic context. The primary functions which distinguish money are as ...
held by the public, ''R'' is the rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
on an alternative, higher yielding asset in excess of that on money, ''Y'' is the public's real income
Real income is the income of individuals or nations after adjusting for inflation. It is calculated by dividing nominal income by the price level. Real variables such as real income and real GDP are variables that are measured in physical units, ...
, ''U'' is an error term assumed to be lognormally distributed, ''A'' is a scale parameter to be estimated, and ''b'' and ''c'' are elasticity parameters to be estimated. Taking logs yields
:
where ''m'' = log ''M'', ''a'' = log ''A'', ''r'' = log ''R'', ''y'' = log ''Y'', and ''u'' = log ''U'' with ''u'' being normally distributed. This equation can be estimated using ordinary least squares
In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model (with fixed level-one effects of a linear function of a set of explanatory variables) by the prin ...
.
Another economic example is the estimation of a firm's Cobb–Douglas production function
In economics and econometrics, the Cobb–Douglas production function is a particular functional form of the production function, widely used to represent the technological relationship between the amounts of two or more inputs (particularly phy ...
, which is the right side of the equation
:
in which ''Q'' is the quantity of output that can be produced per month, ''N'' is the number of hours of labor employed in production per month, ''K'' is the number of hours of physical capital utilized per month, ''U'' is an error term assumed to be lognormally distributed, and ''A'', , and are parameters to be estimated. Taking logs gives the linear regression equation
:
where ''q'' = log ''Q'', ''a'' = log ''A'', ''n'' = log ''N'', ''k'' = log ''K'', and ''u'' = log ''U''.
Log–log regression can also be used to estimate the fractal dimension
In mathematics, more specifically in fractal geometry, a fractal dimension is a ratio providing a statistical index of complexity comparing how detail in a pattern (strictly speaking, a fractal pattern) changes with the scale at which it is me ...
of a naturally occurring fractal
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illu ...
.
However, going in the other direction – observing that data appears as an approximate line on a log–log scale and concluding that the data follows a power law – is not always valid.
In fact, many other functional forms appear approximately linear on the log–log scale, and simply evaluating the goodness of fit
The goodness of fit of a statistical model describes how well it fits a set of observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Such measure ...
of a linear regression
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables (also known as dependent and independent variables). The case of one explanatory variable is call ...
on logged data using the coefficient of determination
In statistics, the coefficient of determination, denoted ''R''2 or ''r''2 and pronounced "R squared", is the proportion of the variation in the dependent variable that is predictable from the independent variable(s).
It is a statistic used i ...
(''R''2) may be invalid, as the assumptions of the linear regression model, such as Gaussian error, may not be satisfied; in addition, tests of fit of the log–log form may exhibit low statistical power
In statistics, the power of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H_0) when a specific alternative hypothesis (H_1) is true. It is commonly denoted by 1-\beta, and represents the chances ...
, as these tests may have low likelihood of rejecting power laws in the presence of other true functional forms. While simple log–log plots may be instructive in detecting possible power laws, and have been used dating back to Pareto in the 1890s, validation as a power laws requires more sophisticated statistics.[
These graphs are also extremely useful when data are gathered by varying the control variable along an exponential function, in which case the control variable ''x'' is more naturally represented on a log scale, so that the data points are evenly spaced, rather than compressed at the low end. The output variable ''y'' can either be represented linearly, yielding a ]lin–log 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 v ...
(log ''x'', ''y''), or its logarithm can also be taken, yielding the log–log graph (log ''x'', log ''y'').
Bode plot (a graph
Graph may refer to:
Mathematics
*Graph (discrete mathematics), a structure made of vertices and edges
**Graph theory, the study of such graphs and their properties
*Graph (topology), a topological space resembling a graph in the sense of discre ...
of the frequency response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of sy ...
of a system) is also log–log plot.
See also
* Semi-log plot
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 o ...
(lin–log or log–lin)
* Power law
In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...
* Zipf law
References
External links
Non-Newtonian calculus website
{{DEFAULTSORT:Log-Log Graph
Logarithmic scales of measurement
Statistical charts and diagrams
Non-Newtonian calculus
de:Logarithmenpapier#Doppeltlogarithmisches Papier