Content deleted Content added
m Removing link(s): Wikipedia:Articles for deletion/Multiplicative calculus closed as delete (XFDcloser) |
mNo edit summary |
||
Line 3:
In [[science]] and [[engineering]], a '''log–log graph''' or '''log–log plot''' is a two-dimensional graph of numerical data that uses [[logarithmic scale]]s on both the horizontal and vertical axes. [[Monomial]]s – relationships of the form <math>y=ax^k</math> – appear as straight lines in a log–log graph, with the power term corresponding to the slope, and the constant term corresponding to the intercept of the line. 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 <math>y=ax^k,</math> taking the logarithm of the equation (with any base) yields:
:<math>\log y = k \log x + \log a.</math>
Line 11:
== Equations ==
The equation for a line on a log–log scale would be:
Line 19 ⟶ 18:
where ''m'' is the slope and ''b'' is the intercept point on the log plot.
=== Slope of a log–log plot ===
[[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''<sub>1</sub> and ''x''<sub>2</sub>. Using the above equation:
Line 38 ⟶ 36:
:<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''<sub>0</sub>, ''F''<sub>0</sub>), where ''F''<sub>0</sub> is shorthand for ''F''(''x''<sub>0</sub>), somewhere on the straight line in the above graph, and further some other ''arbitrary point'' (''x''<sub>1</sub>, ''F''<sub>1</sub>) on the same graph. Then from the slope formula above:
Line 69 ⟶ 66:
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
Line 104 ⟶ 100:
:<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 [[
:<math>m_t = a + br_t + cy_t + u_t,</math>
Line 128 ⟶ 124:
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]] (a [[plot (graphics)|graph]] of the [[frequency response]] of a system) is also
== See also ==
* [[Semi-log plot]] (
== External links ==
* [https://sites.google.com/site/nonnewtoniancalculus/ Non-Newtonian calculus website]
| |||