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.

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):

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

[edit] Generalization

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

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