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:

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

[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

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