Logarithmic mean

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

$$L(x,y)=\{\begin{array}{cc}x,& \text{if }x=y,\\ \frac{x-y}{\ln x-\ln y},& \text{otherwise},\end{array}$$

for $x,y\in ℝ$, such that $x,y>0$.

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. More precisely, for $p,x,y\in ℝ$ with $x\ne y$ and $p>1$, we have $$\frac{2xy}{x+y}<\sqrt{xy}<\frac{x-y}{\ln x-\ln y}<\frac{x+y}{2}<{\left(\frac{x^p+y^p}{2}\right)}^{1/p},$$ 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 $p$.

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value ξ in the interval between x and y where the derivative f ′ equals the slope of the secant line: $$\mathrm{\exists }\xi \in (x,y):f^{\prime }(\xi )=\frac{f(x)-f(y)}{x-y}$$

The logarithmic mean is obtained as the value of ξ by substituting ln for f and similarly for its corresponding derivative: $$\frac{1}{\xi }=\frac{\ln x-\ln y}{x-y}$$

and solving for ξ: $$\xi =\frac{x-y}{\ln x-\ln y}$$

Integration

The logarithmic mean is also given by the integral $$L(x,y)={\int }_0^1{x}^{1-t}{y}^t\mathrm{d}t.$$

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 x and y.

Two other useful integral representations are$$\frac{1}{L(x,y)}={\int }_0^1\frac{\mathrm{d}t}{tx+(1-t)y}$$and$$\frac{1}{L(x,y)}={\int }_0^{\mathrm{\infty }}\frac{\mathrm{d}t}{(t+x)(t+y)}.$$

Generalization

Mean value theorem of differential calculus

One 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.

We obtain $$L_{\text{MV}}(x_0,\dots ,x_n)=\sqrt[-n]{(-1{)}^{n+1}n\ln \left([x_0,\dots ,x_n]\right)}$$ where $\ln \left([x_0,\dots ,x_n]\right)$ denotes a divided difference of the logarithm. For n = 2 this leads to $$L_{\text{MV}}(x,y,z)=\sqrt{\frac{(x-y)(y-z)(z-x)}{2((y-z)\ln x+(z-x)\ln y+(x-y)\ln z)}}.$$

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex $S$ with $$S=\{({\alpha }_0,\dots ,{\alpha }_n)\in {ℝ}^{n+1};{\alpha }_0+\dots +{\alpha }_n=1\text{ and }{\alpha }_j\ge 0,\text{ for }j=0,\dots ,n\}$$ and an appropriate measure $\mathrm{d}\alpha$ which assigns the simplex a volume of 1, we obtain $$L_{\text{I}}(x_0,\dots ,x_n)={\int }_S{x}_0^{{\alpha }_0}\cdots x_n^{{\alpha }_n}\mathrm{d}\alpha .$$

This can be expressed as the divided differences of the exponential function by $$L_{\text{I}}(x_0,\dots ,x_n)=n!\exp [\ln \left(x_0\right),\dots ,\ln \left(x_n\right)].$$ In the case of n = 2, it is $$L_{\text{I}}(x,y,z)=-2\frac{x(\ln y-\ln z)+y(\ln z-\ln x)+z(\ln x-\ln y)}{(\ln x-\ln y)(\ln y-\ln z)(\ln z-\ln x)}.$$

Connection to other means

Some other means can be expressed in terms of the logarithmic mean.

Other means expressed in terms of the logarithmic mean

Name	Mean	Expression
Arithmetic mean	$\frac{x+y}{2}$	$\frac{L(x^2,y^2)}{L(x,y)}$
Geometric mean	$\sqrt{xy}$	$\sqrt{\frac{L(x,y)}{L(\frac{1}{x},\frac{1}{y})}}$
Harmonic mean	$\frac{2}{\frac{1}{x}+\frac{1}{y}}$	$\frac{L(\frac{1}{x},\frac{1}{y})}{L(\frac{1}{x^2},\frac{1}{y^2})}$

See also

• The logarithmic mean is a special case of the Stolarsky mean.
• Logarithmic mean temperature difference
• Log semiring

References

Citations

Bibliography

• Oilfield Glossary: Term 'logarithmic mean'
• Stolarsky, Kenneth B. (1975). "Generalizations of the Logarithmic Mean". Mathematics Magazine. 48 (2): 87–92. doi:10.2307/2689825. ISSN 0025-570X. JSTOR 2689825.