Logarithmic mean

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):


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

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 $$\xi$$ by substituting $$\ln$$ for $$f$$
 * $$ \frac{1}{\xi} = \frac{\ln x - \ln y}{x-y} $$

and solving for $$\xi$$.
 * $$ \xi = \frac{x-y}{\ln x - \ln y} $$

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 $$L(c\cdot x, c\cdot y) = c\cdot L(x,y)$$.

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