Definition
The logarithmic mean of two positive real numbers $a$ and $b$ (with $a
eq b$) is defined as
$$ L(a,b)=\frac{b-a}{\ln b-\ln a}, $$
where $\ln$ denotes the natural logarithm. For the degenerate case $a = b$, the logarithmic mean is defined by continuity as $L(a,a)=a$.
Overview
The logarithmic mean arises in various branches of mathematics, physics, and engineering, particularly in contexts where a quantity varies exponentially between two limits. It provides a more accurate average than the arithmetic mean when dealing with processes governed by logarithmic relationships, such as heat transfer across a temperature gradient, fluid flow in pipes, and chemical reaction rates. In heat exchanger analysis, the logarithmic mean temperature difference (LMTD) is a fundamental parameter for calculating heat transfer rates.
Etymology/Origin
The term combines “logarithmic,” referring to the natural logarithm function, with “mean,” indicating an average value. The concept dates back to early 20th‑century thermodynamic and heat‑transfer literature, where engineers required an average temperature difference that correctly accounts for exponential temperature profiles. The precise formulation is credited to H. K. M. Kramers and later popularized in standard heat‑transfer textbooks.
Characteristics
- Symmetry: $L(a,b)=L(b,a)$; the order of the arguments does not affect the result.
- Monotonicity: If $a<b$, then $a<L(a,b)<b$; the logarithmic mean always lies strictly between its two arguments.
- Homogeneity: For any positive scalar $\lambda$, $L(\lambda a,\lambda b)=\lambda L(a,b)$.
- Limit behavior: $\displaystyle\lim_{b\to a}L(a,b)=a$, ensuring continuity at the diagonal $a=b$.
- Relation to other means: For any $a,b>0$,
$$ \text{geometric mean } G(a,b)=\sqrt{ab}\le L(a,b)\le \frac{a+b}{2}\text{ (arithmetic mean)}, $$
with equality only when $a=b$.
- Integral representation:
$$ L(a,b)=\exp!\left(\frac{1}{b-a}\int_{a}^{b}\ln x,dx\right), $$
showing that the logarithmic mean is the exponential of the average of the logarithms over the interval $[a,b]$.
Related Topics
- Arithmetic mean, Geometric mean, Harmonic mean – other classical averages.
- Logarithmic mean temperature difference (LMTD) – a key concept in heat‑exchanger design.
- Mean value theorems – mathematical foundations underlying various mean definitions.
- Exponential and logarithmic functions – functions central to the definition of the logarithmic mean.
- Thermodynamics and heat transfer – fields where the logarithmic mean is frequently applied.
- Weighted means – generalizations that assign different weights to values, of which the logarithmic mean is a specific case when the weighting arises from logarithmic scaling.