{{Short description|Difference of two numbers divided by the logarithm of their quotient}} {{distinguish|text=the log-average formulation of the geometric mean or the mean in the log semiring}}
In mathematics, the '''logarithmic mean''' is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.
==Definition== The logarithmic mean is defined by
<math display="block"> L(x, y) = \left \{ \begin{array}{l l} x, & \text{if }x = y,\\ \dfrac{x - y}{\ln x - \ln y}, & \text{otherwise}, \end{array} \right . </math>
for <math>x, y \in \mathbb{R}</math>, such that <math>x, y > 0</math>.
== Inequalities ==
The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal.<ref>{{cite journal | author=B. C. Carlson | title=Some inequalities for hypergeometric functions | journal=Proc. Amer. Math. Soc. | volume=17 | year=1966 | pages=32–39 | doi=10.1090/s0002-9939-1966-0188497-6 | doi-access=free }}</ref><ref> {{cite journal | author1=B. Ostle | author2=H. L. Terwilliger | name-list-style=amp | title=A comparison of two means | journal=Proc. Montana Acad. Sci. | volume=17 | year=1957 | pages=69–70 }}</ref><ref> {{cite journal | author1=Tung-Po Lin | title=The Power Mean and the Logarithmic Mean | journal=The American Mathematical Monthly | year=1974 | volume=81| issue=8 | pages=879–883 | doi=10.1080/00029890.1974.11993684 }}</ref><ref> {{cite journal | author1=Frank Burk | title=The Geometric, Logarithmic, and Arithmetic Mean Inequality | journal=The American Mathematical Monthly | year=1987 | volume=94| issue=6 | pages=527–528 | doi=10.2307/2322844 | jstor=2322844 }}</ref> More precisely, for <math>p, x, y \in \mathbb{R}</math> with <math>x \neq y</math> and <math>p > 1</math>, we have <math display="block"> \frac{2xy}{x + y} < \sqrt{x y} < \frac{x - y}{\ln x - \ln y} < \frac{x + y}{2} < \left(\frac{x^p + y^p}2\right)^{1/p}, </math> where the expressions in the chain of inequalities are, in order: the harmonic mean, the geometric mean, the logarithmic mean, the arithmetic mean, and the generalized arithmetic mean with exponent <math>p</math>.
== Derivation ==
=== Mean value theorem of differential calculus ===
From the mean value theorem, there exists a value {{mvar|ξ}} in the interval between {{mvar|x}} and {{mvar|y}} where the derivative {{mvar|f ′}} equals the slope of the secant line: <math display="block">\exists \xi \in (x, y): \ f'(\xi) = \frac{f(x) - f(y)}{x - y}</math>
The logarithmic mean is obtained as the value of {{mvar|ξ}} by substituting {{math|ln}} for {{mvar|f}} and similarly for its corresponding derivative: <math display="block">\frac{1}{\xi} = \frac{\ln x - \ln y}{x-y}</math>
and solving for {{mvar|ξ}}: <math display="block">\xi = \frac{x-y}{\ln x - \ln y}</math>
=== Integration ===
The logarithmic mean is also given by the integral <math display=block> L(x, y) = \int_0^1 x^{1-t} y^t\,\mathrm{d}t. </math>
This interpretation allows the derivation of some properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by {{mvar|x}} and {{mvar|y}}.
Two other useful integral representations are<math display="block">{1 \over L(x,y)} = \int_0^1 {\operatorname{d}\!t \over t x + (1-t)y}</math>and<math display="block">{1 \over L(x,y)} = \int_0^\infty {\operatorname{d}\!t \over (t+x)\,(t+y)}.</math>
== Generalization ==
=== Mean value theorem of differential calculus ===
One can generalize the mean to {{math|''n'' + 1}} variables by considering the mean value theorem for divided differences for the {{mvar|n}}-th derivative of the logarithm.
We obtain <math display="block">L_\text{MV}(x_0,\, \dots,\, x_n) = \sqrt[-n]{(-1)^{n+1} n \ln\left(\left[x_0,\, \dots,\, x_n\right]\right)}</math> where <math>\ln\left(\left[x_0,\, \dots,\, x_n\right]\right)</math> denotes a divided difference of the logarithm. For {{math|1=''n'' = 2}} this leads to <math display="block">L_\text{MV}(x, y, z) = \sqrt{\frac{(x-y)(y-z)(z-x)}{2 \bigl((y-z) \ln x + (z-x) \ln y + (x-y) \ln z \bigr)}}.</math>
=== Integral ===
The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex <math display="inline">S</math> with <math display="block">S = \{\left(\alpha_0,\dots,\alpha_n\right) \in \mathbb{R}^{n + 1}\,;\, \alpha_0 + \dots + \alpha_n = 1 \text{ and } \alpha_j \ge 0, \text{ for } j = 0, \dots, n\}</math> and an appropriate measure <math display="inline">\mathrm{d}\alpha</math> which assigns the simplex a volume of 1, we obtain <math display="block">L_\text{I}\left(x_0,\dots,x_n\right) = \int_S x_0^{\alpha_0} \cdots x_n^{\alpha_n}\,\mathrm{d}\alpha.</math>
This can be expressed as the divided differences of the exponential function by <math display="block">L_\text{I}\left(x_0,\dots,x_n\right) = n! \exp\left[\ln\left(x_0\right), \dots, \ln\left(x_n\right)\right].</math> In the case of {{math|1=''n'' = 2}}, it is <math display="block">L_\text{I}(x, y, z) = -2 \frac{x(\ln y - \ln z) + y(\ln z - \ln x) + z(\ln x - \ln y)} {(\ln x - \ln y)(\ln y - \ln z)(\ln z - \ln x)}.</math>
== Connection to other means ==
Some other means can be expressed in terms of the logarithmic mean.
{| class="wikitable" style="text-align: center; |+ Other means expressed in terms of the logarithmic mean |- ! Name ! Mean ! Expression |- | Arithmetic mean | <math>\frac{x + y}{2}</math> | <math>\frac{L\left(x^2, y^2\right)}{L(x, y)}</math> |- | Geometric mean | <math>\sqrt{x y}</math> | <math>\sqrt{\frac{L\left(x, y\right)}{L\left( \frac{1}{x}, \frac{1}{y} \right)}}</math> |- | Harmonic mean | <math>\frac{2}{\frac{1}{x}+\frac{1}{y}}</math> | <math>\frac{ L\left( \frac{1}{x}, \frac{1}{y} \right) }{L\left( \frac{1}{x^2}, \frac{1}{y^2} \right)}</math> |}
== See also == * The logarithmic mean is a special case of the Stolarsky mean. * Logarithmic mean temperature difference * Log semiring
== References == ;Citations {{Reflist}} ;Bibliography *[https://web.archive.org/web/20060215011645/http://jipam-old.vu.edu.au/v4n4/088_03.html Oilfield Glossary: Term 'logarithmic mean'] * {{mathworld|Arithmetic-Logarithmic-GeometricMeanInequality|Arithmetic-Logarithmic-Geometric-Mean Inequality}} * {{Cite journal |last=Stolarsky |first=Kenneth B. |date=1975 |title=Generalizations of the Logarithmic Mean |url=https://www.jstor.org/stable/2689825 |journal=Mathematics Magazine |volume=48 |issue=2 |pages=87–92 |doi=10.2307/2689825 |jstor=2689825 |issn=0025-570X|url-access=subscription }}
Mean Category:Means