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In
science Science is a systematic discipline that builds and organises knowledge in the form of testable hypotheses and predictions about the universe. Modern science is typically divided into twoor threemajor branches: the natural sciences, which stu ...
and
engineering Engineering is the practice of using natural science, mathematics, and the engineering design process to Problem solving#Engineering, solve problems within technology, increase efficiency and productivity, and improve Systems engineering, s ...
, 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 method used to display numerical data that spans a broad range of values, especially when there are significant differences among the magnitudes of the numbers involved. Unlike a linear Scale (measurement) ...
s on both the horizontal and vertical axes. Power functions – relationships of the form 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=ax^k, taking the logarithm of the equation (with any base) yields: \log y = k \log x + \log a. Setting X = \log x and Y = \log y, which corresponds to using a log–log graph, yields the equation Y = mX + b where ''m'' = ''k'' is the slope of the line (
gradient In vector calculus, the gradient of a scalar-valued differentiable function f of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p gives the direction and the rate of fastest increase. The g ...
) 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_F(x) = m \log_x + b, F(x) = x^m\cdot10^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 ''x''1 and ''x''2. Using the below equation: \log (x_1)= m \log (x_1) + b, and \log (x_2)= m \log(x_2) + b. The slope ''m'' is found taking the difference: m = \frac = \frac , 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: \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'' (''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: m = \frac which leads to \log(F_1 / F_0) = m \log(x_1 / x_0) = \log x_1 / x_0)^m Notice that 10log10(''F''1) = ''F''1. Therefore, the logs can be inverted to find: \frac = \left(\frac\right)^m or F_1 = \frac \, x^m, which means that F(x) = \mathrm\cdot x^m. 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 (''x''0, ''F''0) and (''x''1, ''F''1) will have the function: F(x) = \left(\frac \right)^\frac , Of course, the inverse is true too: any function of the form F(x) = \mathrm \cdot x^m 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 F(x) = \mathrm\cdot x^m. and integrate it. Since it is only operating on a definite integral (two defined endpoints), the area A under the plot takes the form A(x) = \int_^ F(x) \, dx = \left.\frac \cdot x^\_^ Rearranging the original equation and plugging in the fixed point values, it is found that \mathrm = \frac Substituting back into the integral, you find that for ''A'' over ''x''0 to ''x''1 \begin A &= \frac\cdot (x_1^-x_0^) \\ .2ex\log A &= \log \left frac \cdot (x_1^-x_0^)\right\\ &= \log \frac - \log \frac + \log (x_1^-x_0^) \\ &= \log \frac + \log \left(\frac\right) \\ &= \log \frac + \log \left(\frac\cdot x_1 - \frac\right) \end Therefore, A = \frac \cdot \left _1 \cdot \left(\frac \right)^m - x_0\right For ''m'' = −1, the integral becomes \begin A_ &= \int_^ F(x) \, dx = \int_^ \frac \, dx = \frac \int_^ \frac = F_0 \cdot x_0 \cdot \Big, _^ \\ A_ &= F_0 \cdot x_0 \cdot \ln \frac \end


Log-log linear regression models

Log–log plots are often use for visualizing log-log linear regression models with (roughly)
log-normal In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
, or Log-logistic, errors. In such models, after log-transforming the dependent and independent variables, a
Simple linear regression In statistics, simple linear regression (SLR) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x ...
model can be fitted, with the errors becoming
homoscedastic In statistics, a sequence of random variables is homoscedastic () if all its random variables have the same finite variance; this is also known as homogeneity of variance. The complementary notion is called heteroscedasticity, also known as hete ...
. This model is useful when dealing with data that exhibits exponential growth or decay, while the errors continue to grow as the independent value grows (i.e., heteroscedastic error). As above, in a log-log linear model the relationship between the variables is expressed as a power law. Every unit change in the independent variable will result in a constant percentage change in the dependent variable. The model is expressed as: :y = a \cdot x^b \cdot e^\epsilon Taking the logarithm of both sides, we get: :\log(y) = \log(a) + b \cdot \log(x) + \epsilon This is a
linear equation In mathematics, a linear equation is an equation that may be put in the form a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coeffici ...
in the logarithms of x and y, with \log(a) as the intercept and b as the slope. In which \epsilon \sim \textrm(\mu, \sigma^2), and e^\epsilon \sim \textrm(\mu, \sigma^2). Figure 1 illustrates how this looks. It presents two plots generated using 10,000 simulated points. The left plot, titled 'Concave Line with Log-Normal Noise', displays a
scatter plot A scatter plot, also called a scatterplot, scatter graph, scatter chart, scattergram, or scatter diagram, is a type of plot or mathematical diagram using Cartesian coordinates to display values for typically two variables for a set of dat ...
of the observed data (y) against the independent variable (x). The red line represents the 'Median line', while the blue line is the 'Mean line'. This plot illustrates a dataset with a power-law relationship between the variables, represented by a concave line. When both variables are log-transformed, as shown in the right plot of Figure 1, titled 'Log-Log Linear Line with Normal Noise', the relationship becomes linear. This plot also displays a scatter plot of the observed data against the independent variable, but after both axes are on a logarithmic scale. Here, both the mean and median lines are the same (red) line. This transformation allows us to fit a
Simple linear regression In statistics, simple linear regression (SLR) is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable (conventionally, the ''x ...
model (which can then be transformed back to the original scale - as the median line). The transformation from the left plot to the right plot in Figure 1 also demonstrates the effect of the log transformation on the distribution of noise in the data. In the left plot, the noise appears to follow a
log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
, which is right-skewed and can be difficult to work with. In the right plot, after the log transformation, the noise appears to follow a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, which is easier to reason about and model. This normalization of noise is further analyzed in Figure 2, which presents a line plot of three error metrics (
Mean Absolute Error In statistics, mean absolute error (MAE) is a measure of Error (statistics), errors between paired observations expressing the same phenomenon. Examples of ''Y'' versus ''X'' include comparisons of predicted versus observed, subsequent time vers ...
- MAE, Root Mean Square Error - RMSE, and Mean Absolute Logarithmic Error - MALE) calculated over a sliding window of size 28 on the x-axis. The y-axis gives the error, plotted against the independent variable (x). Each error metric is represented by a different color, with the corresponding smoothed line overlaying the original line (since this is just simulated data, the error estimation is a bit jumpy). These error metrics provide a measure of the noise as it varies across different x values. Log-log linear models are widely used in various fields, including economics, biology, and physics, where many phenomena exhibit power-law behavior. They are also useful in regression analysis when dealing with heteroscedastic data, as the log transformation can help to stabilize the variance.


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 a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and interac ...
. 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 (economics), M1 (dire ...
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: it ...
, in which it can be assumed that money demand at time ''t'' is given by M_t = AR_t^bY_t^cU_t, 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: m ...
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 over a specified time period, such as i ...
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 ...
, ''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 m_t = a + br_t + cy_t + u_t, where ''m'' = log ''M'', ''a'' = log ''A'', ''r'' = log ''R'', ''y'' = log ''Y'', and ''u'' = log ''U'' with ''u'' being
normally distributed In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real number, real-valued random variable. The general form of its probability density function is f(x ...
. 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 In statistics, linear regression is a statistical model, model that estimates the relationship ...
. 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 Q_t=AN_t^K_t^U_t, 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'', \alpha, and \beta are parameters to be estimated. Taking logs gives the linear regression equation q_t = a + \alpha n_t + \beta k_t + u_t 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, a fractal dimension is a term invoked in the science of geometry to provide a rational statistical index of complexity detail in a pattern. A fractal pattern changes with the Scaling (geometry), scale at which it is measured. It ...
of a naturally occurring
fractal In mathematics, a fractal is a Shape, 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 scale ...
. 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 measur ...
of a
linear regression In statistics, linear regression is a statistical model, model that estimates the relationship between a Scalar (mathematics), scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable). A mode ...
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 in t ...
(''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 frequentist statistics, power is the probability of detecting a given effect (if that effect actually exists) using a given test in a given context. In typical use, it is a function of the specific test that is used (including the choice of tes ...
, 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 (log ''x'', ''y''), or its logarithm can also be taken, yielding the log–log graph (log ''x'', log ''y'').
Bode plot In electrical engineering and control theory, a Bode plot is a graph of the frequency response of a system. It is usually a combination of a Bode magnitude plot, expressing the magnitude (usually in decibels) of the frequency response, and a B ...
(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 discret ...
of the
frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and Phase (waves), phase of the output as a function of input frequency. The frequency response is widely used in the design and ...
of a system) is also log–log plot. In
chemical kinetics Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is different from chemical thermodynamics, which deals with the direction in which a ...
, the general form of the dependence of the
reaction rate The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per u ...
on concentration takes the form of a power law (
law of mass action In chemistry, the law of mass action is the proposition that the rate of a chemical reaction is directly proportional to the product of the activities or concentrations of the reactants. It explains and predicts behaviors of solutions in dy ...
), so a log-log plot is useful for estimating the reaction parameters from experiment.


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 of ...
(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 relative change in the other quantity proportional to the ...
* Zipf law * Log-linear model *
Log-normal distribution In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normal distribution, normally distributed. Thus, if the random variable is log-normally distributed ...
* Log-logistic distribution *
Data transformation (statistics) In statistics, data transformation is the application of a deterministic mathematical function to each point in a data set—that is, each data point ''zi'' is replaced with the transformed value ''yi'' = ''f''(''zi''), where ''f'' is a functi ...
*
Variance-stabilizing transformation In applied statistics, a variance-stabilizing transformation is a data transformation that is specifically chosen either to simplify considerations in graphical exploratory data analysis or to allow the application of simple regression-based or ana ...


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