Log–log plot: Difference between revisions

Revision as of 03:19, 16 November 2022

Revision as of 23:18, 18 March 2023

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== Equations ==

The equation for a line on a log–log scale would be:

:<math display="block"> \log[log_{10}F (x_1x)] = m \log (x_1)log_{10}x + b, </math>

:<math display="block"> \log_{10}F(x) = x^m \log_{10}x + cdot10^b, </math>

:<math> F(x) = x^m\cdot10^b, </math>

where ''m'' is the slope and ''b'' is the intercept point on the log plot.

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[[Image:Slope of log-log plot.PNG|thumbnail|250px|Finding the slope of a log–log plot using ratios]]

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:

:<math display="block"> \log[F (x_2x_1)] = m \log (x_2x_1) + b., </math>

:<math> \log[F (x_1)] = m \log (x_1) + b, </math>

and

:<math display="block"> \log[F (x_1/x_2)] = -m \log(x_2/x_1) + b. </math>

:<math> \log[F (x_2)] = m \log(x_2) + b. </math>

The slope ''m'' is found taking the difference:

:<math display="block"> 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)}, </math>

:<math> 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)}, </math>

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:

:<math> \log(F_1 / F_0) display= m"block"> \log(x_1 / x_0x_2) = -\log[(x_2/x_1 / x_0)^m ]. </math>

:<math> \log(x_1/x_2) = -\log(x_2/x_1). </math>

=== 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:

:<math display="block"> m = \frac {\log (F_1 / F_0)}{\log(x_1 / x_0)} </math>

:<math> m = \frac {\log (F_1 / F_0)}{\log(x_1 / x_0)} </math>

which leads to

<math display="block"> \log(F_1 / F_0) = m \log(x_1 / x_0) = \log[(x_1 / x_0)^m ]. </math>

:<math> \log(F_1 / F_0) = m \log(x_1 / x_0) = \log[(x_1 / x_0)^m ]. </math>

Notice that 10^{log₁₀(''F''₁)} = ''F''₁. Therefore, the logs can be inverted to find:

: <math display="block"> \frac{F_1}{F_0} = \left(\frac{x_1}{x_0}\right)^m </math>

: <math> \frac{F_1}{F_0} = \left(\frac{x_1}{x_0}\right)^m </math>

or

: <math display="block">F_1 = \frac{F_0}{x_0^m} \, x^m, </math>

: <math>F_1 = \frac{F_0}{x_0^m} \, x^m, </math>

which means that

: <math display="block"> F(x) = \mathrm{constant}\cdot x^m. </math>

: <math> F(x) = \mathrm{constant}\cdot x^m. </math>

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,&nbsp;''x''₀) and (''F''₁,&nbsp;''x''₁) will have the function:

:<math display="block"> F(x) = {F_0}\left(\frac{x}{x_0} \right)^\frac {\log (F_1/F_0)}{\log(x_1/x_0)}, </math>

Of course, the inverse is true too: any function of the form

:<math display="block"> F(x) = \mathrm{constant} \cdot x^m</math>

:<math> F(x) = \mathrm{constant} \cdot x^m</math>

will have a straight line as its log–log graph representation, where the slope of the line is&nbsp;''m''.

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Rearranging the original equation and plugging in the fixed point values, it is found that

<math display="block"> \mathrm{constant} = \frac{F_0}{x_0^m} </math>

Substituting back into the integral, you find that for ''A'' over ''x''₀ to ''x''₁

<math display="block">\begin{align}

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\end{align}</math>

Therefore:, <math> A = \frac{F_0}{m+1} \cdot \left[x_1 \cdot \left(\frac {x_1}{x_0}\right)^m - x_0\right] </math>

For ''m''&nbsp;=&nbsp;−1, the integral becomes

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One example is the estimation of [[money demand]] functions based on [[Money demand#Inventory theory|inventory theory]], in which it can be assumed that money demand at time ''t'' is given by

:<math display="block">M_t = AR_t^bY_t^cU_t,</math>

:<math>M_t = AR_t^bY_t^cU_t,</math>

where ''M'' is the real quantity of [[money]] held by the public, ''R'' is the [[rate of return]] on an alternative, higher yielding asset in excess of that on money, ''Y'' is the public's [[real income]], ''U'' is an error term assumed to be [[log-normal distribution|lognormally distributed]], ''A'' is a scale parameter to be estimated, and ''b'' and ''c'' are [[Elasticity (economics)|elasticity]] parameters to be estimated. Taking logs yields

:<math display="block">m_t = a + br_t + cy_t + u_t,</math>

:<math>m_t = a + br_t + cy_t + u_t,</math>

where ''m'' = log ''M'', ''a'' = log ''A'', ''r'' = log ''R'', ''y'' = log ''Y'', and ''u'' = log ''U'' with ''u'' being [[normal distribution|normally distributed]]. This equation can be estimated using [[ordinary least squares]].

Another economic example is the estimation of a firm's [[Cobb–Douglas production function]], which is the right side of the equation

:<math display="block">Q_t=AN_t^{\alpha}K_t^{\beta}U_t,</math>

:<math>Q_t=AN_t^{\alpha}K_t^{\beta}U_t,</math>

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'', <math>\alpha</math>, and <math>\beta</math> are parameters to be estimated. Taking logs gives the linear regression equation

:<math display="block">q_t = a + \alpha n_t + \beta k_t + u_t</math>

:<math>q_t = a + \alpha n_t + \beta k_t + u_t</math>

where ''q'' = log ''Q'', ''a'' = log ''A'', ''n'' = log ''N'', ''k'' = log ''K'', and ''u'' = log ''U''.