Logarithmic mean

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In mathematics, the logarithmic mean is a function of two numbers which is equal to their difference divided by the logarithm of their quotient. In symbols:

$\begin{matrix} M_{\mbox{lm}}(x,y) &= \lim_{(\xi,\eta)\to(x,y)} \frac{\eta - \xi}{\ln \eta - \ln \xi} \\ &= \begin{cases} x & \mbox{if }x=y \\ \frac{y - x}{\ln y - \ln x} & \mbox{else} \end{cases} \end{matrix}$

for the positive numbers $x, y$ . This measure is useful in engineering problems involving heat and mass transfer.

Inequalities

The logarithmic mean of two numbers is smaller than the arithmetic mean but larger than the geometric mean (unless the numbers are the same of course, in which case all three means are equal to the numbers):

$\forall x>0\ \forall y>0\ x\ne y\Rightarrow \sqrt{x\cdot y} < \frac{y - x}{\ln y - \ln x} < \frac{x+y}{2}$

Derivation of the mean

Mean value theorem of differential calculus

From the mean value theorem

$\exists \xi\in[x,y]\ f'(\xi) = \frac{f(x)-f(y)}{x-y}$

the logarithmic mean is obtained as the value of $ξ$ by substituting $ln$ for $f$

$\frac{1}{\xi} = \frac{\ln x - \ln y}{x-y}$

and solving for $ξ$ .

$\xi = \frac{x-y}{\ln x - \ln y}$

Integration

The logarithmic mean can also be interpreted as the area under an exponential curve.

$L(x,y) = \int_0^1 x^{1-t}\cdot y^t\ \mathrm{d}t$

(Check $\begin{array}{rcl} \int_0^1 x^{1-t}\cdot y^t\ \mathrm{d}t &=& \int_0^1 \left(\frac{y}{x}\right)^t\cdot x\ \mathrm{d}t \\ &=& x\cdot \int_0^1 \exp\left(t\cdot\ln \frac{y}{x}\right) \mathrm{d}t \\ &=& \frac{x}{\ln \frac{y}{x}} \cdot \left[ \exp\left(t\cdot\ln \frac{y}{x}\right) \right]_{t=0}^{1} \\ &=& \frac{x}{\ln \frac{y}{x}} \cdot \left(\frac{y}{x}-1\right) \end{array}$ )

The area interpretation allows to easily derive basic properties of the logarithmic mean. Since the exponential function is monotonic the integral over an interval of length 1 is bounded by $x$ and $y$ . The Homogenity of the integral operator is transferred to the mean operator, that is $L(c\cdot x, c\cdot y) = c\cdot L(x,y)$ .

Generalization

Mean value theorem of differential calculus

You can generalize the mean to $n + 1$ variables by considering the mean value theorem for divided differences for the $n$ th derivative of the logarithm. You obtain

$L_{\mathrm{MV}}(x_0,\dots,x_n) = \sqrt[-n]{(-1)^{(n+1)}\cdot n \cdot \ln[x_0,\dots,x_n]}$

where $\ln[x_0,\dots,x_n]$ denotes a divided difference of the logarithm.

For $n = 2$ this leads to

$L_{\mathrm{MV}}(x,y,z) = \sqrt{\frac{(x-y)\cdot(y-z)\cdot(z-x)}{2\cdot((y-z)\cdot\ln x + (z-x)\cdot\ln y + (x-y)\cdot\ln z)}}$ .

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex $S$ with $S = \{(\alpha_0,\dots,\alpha_n) : \alpha_0+\dots+\alpha_n=1\ \land\ \alpha_0\ge0\ \land\ \dots\ \land\ \alpha_n\ge0\}$ and an appropriate measure $dα$ which assigns the simplex a volume of 1, we obtain

$L_{\mathrm{I}}(x_0,\dots,x_n) = \int_S x_0^{\alpha_0}\cdot\dots\cdot x_n^{\alpha_n}\ \mathrm{d}\alpha$

This can be simplified using divided differences of the exponential function to

$L_{\mathrm{I}}(x_0,\dots,x_n) = n!\cdot\exp[\ln x_0, \dots, \ln x_n]$ .

Example $n = 2$

$L_{\mathrm{I}}(x,y,z) = -2\cdot\frac{x\cdot(\ln y-\ln z) + y\cdot(\ln z-\ln x) + z\cdot(\ln x-\ln y)}{(\ln x-\ln y)\cdot(\ln y-\ln z)\cdot(\ln z-\ln x)}$ .

References

Acknowledgement and Attribution Regarding Sources of Content

Some of the initial content on this page may be incorporated in part from copyleft sources in the public domain including wikis such as Wikipedia and AskDrWiki. Drug information for patients came from the The National Library of Medicine. Infectious disease information may have come from the Centers for Disease Control (CDC). Differential Diagnoses are drawn from clinicians as well as an amalgamation of 3 sources: 1.The Disease Database; 2. Kahan, Scott, Smith, Ellen G. In A Page: Signs and Symptoms. Malden, Massachusetts: Blackwell Publishing, 2004:3; 3. Sailer, Christian, Wasner, Susanne. Differential Diagnosis Pocket. Hermosa Beach, CA: Borm Bruckmeir Publishing LLC, 2002:7 .

Logarithmic mean

Contents

Inequalities

Derivation of the mean

Mean value theorem of differential calculus

Integration

Generalization

Mean value theorem of differential calculus

Integral

See also

References

Acknowledgement and Attribution Regarding Sources of Content

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