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Logarithmic mean

The logarithmic mean of two x and y with x \neq y is a type of defined by the formula L(x, y) = \frac{x - y}{\ln x - \ln y}, where \ln denotes the natural logarithm; when x = y, it is defined by as L(x, x) = x. This mean arises naturally in contexts involving or multiplicative processes, providing a value that interpolates between the \sqrt{xy} and the \frac{x + y}{2}. For x > y > 0 and x \neq y, the logarithmic mean satisfies the strict \sqrt{xy} < L(x, y) < \frac{x + y}{2}, positioning it within the hierarchy of classical means; equality holds only in the limit as x approaches y. This property stems from the concavity of the logarithm function and can be derived using integral representations, such as expressing L(x, y) as the average value of a linear function over a logarithmic scale. The logarithmic mean is homogeneous of degree 1, meaning L(tx, ty) = t \cdot L(x, y) for t > 0, and it is symmetric in the sense that L(x, y) = L(y, x), though the formula requires careful handling of the order to ensure positivity. Introduced in generalizations by Kenneth B. Stolarsky in 1975, the logarithmic mean serves as a special case of the broader Stolarsky mean E_{p,q}(x, y), obtained when parameters p = 0 and q = 1, which unifies various through a parameterized form involving ratios of differences and logarithms. Extensions to multiple variables exist, often defined via symmetric sums or iterative pairings, preserving similar chains with power means. Beyond , the logarithmic mean finds practical applications in , notably in the calculation of the (LMTD) for heat exchangers, where it quantifies the effective temperature gradient driving convective under steady-state conditions with constant fluid properties. In , it appears in analyses of log-normal distributions and index numbers, such as the logarithmic mean Divisia index for decomposing aggregate changes in economic data.

Fundamentals

Definition

The logarithmic mean is a function that provides a type of average for two positive real numbers, positioned between the geometric and arithmetic means in the hierarchy of classical means. For distinct positive real numbers x > 0 and y > 0 with x \neq y, the logarithmic mean L(x, y) is defined as L(x, y) = \frac{x - y}{\ln x - \ln y}, where \ln denotes the natural logarithm (base e). When x = y > 0, the expression is indeterminate, but by (or application of to the limiting form), L(x, x) = x. The logarithmic mean is defined only for ; it is undefined if x \leq 0 or y \leq 0, or if x = y = 0, due to the domain of the natural logarithm.

Basic Properties

The logarithmic mean L(x, y) exhibits several fundamental properties that arise directly from its definition. It is symmetric, satisfying L(x, y) = L(y, x) for all x, y > 0. This follows immediately from the form of the expression, as interchanging x and y yields the negative of both numerator and denominator, preserving the value. Additionally, L(x, y) is homogeneous of degree one, meaning L(tx, ty) = t L(x, y) for all t > 0 and x, y > 0. To verify this, substitute into the definition: the numerator becomes t(x - y) and the denominator \ln(tx) - \ln(ty) = \ln t + \ln x - \ln t - \ln y = \ln x - \ln y, so the factor t factors out. The function is continuous on the (0, \infty) \times (0, \infty), including at points where x = y, where it takes the value x (or y) by the standard limiting convention or applied to the . This continuity ensures well-behaved behavior across the positive reals. For fixed x > 0, as y \to 0^+, L(x, y) \to [0](/page/0). This limit is obtained by observing that the numerator x - y \to x > 0 while the denominator \ln x - \ln y \to +\infty (since \ln y \to -\infty), so the ratio approaches . Similarly, for fixed x > 0, as y \to +\infty, L(x, y) \sim \frac{y}{\ln y}. Assuming y > x, rewrite L(x, y) = \frac{y - x}{\ln y - \ln x} = \frac{y(1 - x/y)}{\ln y (1 - \ln x / \ln y)}. As y \to +\infty, x/y \to 0 and \ln x / \ln y \to 0 (since \ln y \to +\infty), yielding the asymptotic equivalence \frac{y(1 - 0)}{\ln y (1 - 0)} = \frac{y}{\ln y}.

Derivations

Mean Value Theorem Approach

The (MVT) states that if a f is continuous on the closed [a, b] and differentiable on the open (a, b), then there exists some \xi \in (a, b) such that f'(\xi) = \frac{f(b) - f(a)}{b - a}. To derive the logarithmic mean L(x, y) for positive real numbers x > y > 0, consider the function f(t) = \ln t, which is strictly increasing and continuous on (0, \infty), and differentiable with derivative f'(t) = 1/t. Applying the MVT to f on the interval [y, x] yields the existence of \xi \in (y, x) such that \frac{1}{\xi} = \frac{\ln x - \ln y}{x - y}, which rearranges to \xi = \frac{x - y}{\ln x - \ln y} = L(x, y). This derivation shows that L(x, y) equals \xi, the intermediate point guaranteed by the MVT where the instantaneous rate of change of \ln t (i.e., $1/t) matches the average rate of change of \ln t over [y, x]. Thus, L(x, y) provides a natural interpretation as the value of t at which the reciprocal of the function equals the secant slope of the logarithm between x and y.

Integral Representation

The logarithmic mean L(x, y) of two positive real numbers x and y (with x \neq y) admits an integral representation as L(x, y) = \int_0^1 x^{1-t} y^t \, dt. This form expresses the mean as the average value of the family of weighted geometric means x^{1-t} y^t over the parameter t \in [0, 1], where t parameterizes the weight on y. To verify this representation from the closed-form expression L(x, y) = \frac{x - y}{\ln x - \ln y}, assume without loss of generality that x > y > 0. Substitute into the integral: \int_0^1 x^{1-t} y^t \, dt = x \int_0^1 \left( \frac{y}{x} \right)^t \, dt. Let r = \ln(y/x) < 0, so \left( y/x \right)^t = e^{t r}. The integral becomes x \int_0^1 e^{t r} \, dt = x \left[ \frac{e^{t r}}{r} \right]_0^1 = x \cdot \frac{e^r - 1}{r} = x \cdot \frac{y/x - 1}{\ln(y/x)} = \frac{x - y}{\ln x - \ln y}, confirming the equivalence via this change of variables. An alternative integral form highlights the connection to the reciprocal of the integrand's inverse: L(x, y) = \frac{x - y}{\int_y^x \frac{1}{u} \, du}, which interprets L(x, y) as the reciprocal of the average value of $1/u over the interval [y, x]. This follows directly from the closed form, since \int_y^x \frac{1}{u} \, du = \ln x - \ln y. The exponential connection arises from viewing the integrand in logarithmic coordinates: x^{1-t} y^t = \exp\left( (1-t) \ln x + t \ln y \right), so the integral averages the exponential of a linear interpolation between \ln x and \ln y. In contrast, \exp\left( \int_0^1 \ln(x^{1-t} y^t) \, dt \right) = \exp\left( \int_0^1 [(1-t) \ln x + t \ln y] \, dt \right) = \sqrt{xy}, the geometric mean, illustrating the distinction between averaging exponentials and exponentiating averages.

Properties and Inequalities

Key Inequalities

For positive real numbers x and y with x \neq y, the L(x, y) satisfies the classical chain of inequalities among the harmonic, geometric, arithmetic, and logarithmic means: \frac{2xy}{x + y} \leq \sqrt{xy} \leq L(x, y) \leq \frac{x + y}{2}, where equality holds in all parts if and only if x = y. The inequality \sqrt{xy} \leq L(x, y) follows from an integral form of the applied to the change of variables a = \ln y and b = \ln x (assuming without loss of generality that x > y > 0). Specifically, \int_a^b e^t \, dt \cdot \int_a^b e^{-t} \, dt \geq (b - a)^2, which simplifies to (x - y)^2 / (xy) \geq [\ln(x/y)]^2, or equivalently L(x, y) \geq \sqrt{xy}. The inequality L(x, y) \leq (x + y)/2 is similarly established using the Cauchy-Schwarz inequality in integral form: \left( \int_y^x 1 \, dt \right)^2 \leq \left( \int_y^x t \, dt \right) \left( \int_y^x \frac{1}{t} \, dt \right), yielding (x - y)^2 \leq [(x^2 - y^2)/2] [\ln(x/y)], which rearranges to L(x, y) \leq (x + y)/2. Equality holds if and only if the integrands are proportional, which occurs precisely when x = y. The preceding parts of the chain, \frac{2xy}{x + y} \leq \sqrt{xy} and \sqrt{xy} \leq (x + y)/2, are the standard harmonic-geometric and geometric-arithmetic mean inequalities, which follow from applying the arithmetic-geometric mean inequality to the reciprocals for the former and directly via on the concave function \ln t for the latter. Additionally, for fixed y > 0, L(x, y) is strictly increasing in x > 0. To see this, normalize by setting u = x/y > 0, so L(x, y)/y = (u - 1)/\ln u for u \neq 1. The derivative of (u - 1)/\ln u is positive for u > 0, u \neq 1, as the numerator \ln u - (u - 1)/u > 0 follows from the strict convexity of -\ln u.

Bounds and Approximations

The logarithmic mean L(x, y) for positive x, y with x \neq y satisfies the fundamental inequality \sqrt{xy} \leq L(x, y) \leq \frac{x + y}{2}. Tighter bounds are given by G^{2/3} A^{1/3} \leq L(x, y) \leq \frac{2}{3} G + \frac{1}{3} A, where G = \sqrt{xy} is the geometric mean and A = \frac{x + y}{2} is the arithmetic mean; these bounds are sharp and improve upon the classical inequality. For computational approximations when x and y are close, consider the case x = y(1 + h) with |h| < 1 and h small. Then L(x, y) = y \frac{h}{\ln(1 + h)}. The Taylor series expansion of \ln(1 + h) around h = 0 is \ln(1 + h) = h - \frac{h^2}{2} + \frac{h^3}{3} - \frac{h^4}{4} + \cdots. Inverting this series yields \frac{h}{\ln(1 + h)} = 1 + \frac{h}{2} - \frac{h^2}{12} + \frac{h^3}{24} - \cdots, so L(x, y) = y \left(1 + \frac{h}{2} - \frac{h^2}{12} + \frac{h^3}{24} - \cdots \right). Truncating at the quadratic term gives the approximation L(x, y) \approx y \left(1 + \frac{h}{2} - \frac{h^2}{12}\right) = \frac{x + y}{2} - \frac{(x - y)^2}{12 y}; the error in this quadratic approximation is of order O(h^3), or O\left(\left(\frac{x - y}{y}\right)^3\right). A symmetric variant, useful for balanced numerical computation, is L(x, y) \approx \frac{x + y}{2} - \frac{(x - y)^2}{6(x + y)}, which coincides with the above to second order when x \approx y and has similar cubic error behavior.

Generalizations

To Multiple Variables

The generalization of the logarithmic mean to multiple variables, specifically to n+1 positive real numbers x_0, x_1, \dots, x_n, arises from applying the to the nth divided difference of the function f(t) = \ln t. According to this approach, there exists some \xi in the convex hull of \{x_0, \dots, x_n\} such that the nth divided difference satisfies f[x_0, \dots, x_n] = f^{(n)}(\xi)/n!. For f(t) = \ln t, the nth derivative is f^{(n)}(t) = (-1)^{n-1} (n-1)! / t^n. The logarithmic mean L_{MV}(x_0, \dots, x_n) is defined as the reciprocal in a manner analogous to the two-variable case, but more precisely, it is obtained as the nth divided difference of the g(u) = e^u evaluated at u_i = \ln x_i, yielding the closed-form expression L_{MV}(x_0, \dots, x_n) = \sum_{k=0}^n x_k \prod_{\substack{j=0 \\ j \neq k}}^n \frac{1}{\ln x_k - \ln x_j} for distinct x_i, with continuity extensions for equal values. This form captures the multi-variable logarithmic mean through the divided difference structure. When the points x_0, x_1, \dots, x_n are equally spaced, say with common difference h, the divided difference simplifies via the connection to finite differences, where the nth divided difference is the nth forward difference divided by h^n n!. In this case, the formula reduces to expressions involving alternating binomial sums adjusted for the proper sign in the forward difference \Delta^n g(u_0) = \sum_{k=0}^n (-1)^{n-k} \binom{n}{k} g(u_0 + k \delta), linking to numerical integration rules such as higher-order trapezoidal approximations. For instance, the two-variable case (n=1) directly recovers the standard logarithmic mean L(x_0, x_1) = (x_1 - x_0)/(\ln x_1 - \ln x_0), illustrating how the multi-variable form encapsulates the mean value theorem derivation for pairs. This multi-variable logarithmic mean retains key properties of its two-variable counterpart, including homogeneity of degree 1—scaling all x_i by a positive constant \lambda results in L_{MV}(\lambda x_0, \dots, \lambda x_n) = \lambda L_{MV}(x_0, \dots, x_n)—and symmetry with respect to permutations of the arguments when interpreted through the underlying divided difference framework, ensuring the mean is invariant under reordering for the general case. These properties make it suitable for extensions in analysis and approximation theory.

Other Extensions

The logarithmic mean can be generalized to multiple positive real numbers using an integral representation over the simplex. For positive x_1, ..., x_n, the generalized logarithmic mean is L(x_1, \dots, x_n) = \int_{E_{n-1}} \exp\left( \sum_{i=1}^n v_i \log x_i \right) \, dv, where E_{n-1} is the standard (n-1)-simplex { v \in \mathbb{R}^n_+ \mid \sum v_i = 1 }, and dv denotes the induced Lebesgue measure (with the integral normalized such that the total measure of the simplex is 1/(n-1)! to yield a probability-like average). This form equals the expected value of \prod x_i^{v_i} under the uniform distribution on the simplex and reduces to the two-variable logarithmic mean when n=2. Equivalently, in ordered integral coordinates, L(x_1, \dots, x_n) = (n-1)! \int_0^1 \int_0^{1-t_1} \cdots \int_0^{1 - t_1 - \cdots - t_{n-2}} x_1^{1 - t_1 - \cdots - t_{n-1}} x_2^{t_1} \cdots x_n^{t_{n-1}} \, dt_{n-1} \cdots dt_1. This integral extension preserves key properties like monotonicity and homogeneity. It also admits a closed-form expression via divided differences: with u_i = \log x_i and f(u) = e^u, L(x_1, \dots, x_n) is the (n-1)th divided difference f[u_1, \dots, u_n], which expands to L(x_1, \dots, x_n) = \sum_{i=1}^n x_i \prod_{j \neq i} \frac{1}{\log(x_i / x_j)} for distinct x_i (with continuity for equal values). Another extension is the Stolarsky mean, a parametric family generalizing the logarithmic mean. For positive x \neq y and real parameters p, q with p \neq q, it is defined as S_{p,q}(x, y) = \left( \frac{q (x^p - y^p)}{p (x^q - y^q)} \right)^{1/(p-q)}. The logarithmic mean arises as the limiting case \lim_{p \to 0} S_{p,1}(x, y) = L(x, y), providing a continuous interpolation between means such as the geometric mean (p = q = 0 limit) and the arithmetic mean (p = 1, q = 0 limit). This family satisfies internality with respect to standard means and has been extended to multiple variables while preserving symmetry and monotonicity. The weighted logarithmic mean extends the two-variable case by incorporating a weight parameter \alpha \in [0,1]. It can be derived via weighted integrals analogous to the unweighted integral representation, effectively biasing the measure on the simplex toward one variable. This form appears in analyses of convexity and refinement of mean inequalities. For multiple variables, weights can be incorporated into the Dirichlet measure on the simplex for a natural generalization. In limit behaviors, the logarithmic mean emerges distinctly within parametric mean families. For instance, while the power mean M_p(x, y) = \left( \frac{x^p + y^p}{2} \right)^{1/p} approaches the geometric mean as p \to 0, the logarithmic mean is recovered as a specific limit in the Stolarsky family (as noted above) or generalized logarithmic means L_p(x, y) = \left( \frac{x^{p+1} - y^{p+1}}{(p+1)(x - y)} \right)^{1/p}, where \lim_{p \to 0} L_p(x, y) = L(x, y). These limits highlight the logarithmic mean's role as an intermediate between the geometric and arithmetic means in hierarchies of symmetric means.

Applications

Heat and Mass Transfer

In counterflow heat exchangers, the logarithmic mean temperature difference (LMTD) serves as the effective driving force for heat transfer, defined as \Delta T_{lm} = \frac{\Delta T_1 - \Delta T_2}{\ln(\Delta T_1 / \Delta T_2)}, where \Delta T_1 and \Delta T_2 represent the temperature differences between the hot and cold fluids at the inlet and outlet ends, respectively. This formulation arises from the integration of the differential heat transfer equation dQ = U \Delta T \, dA across the exchanger area A, assuming a constant overall heat transfer coefficient U and linear fluid temperature profiles along the flow direction, which yields the logarithmic form due to the exponential nature of the temperature decay. The LMTD offers greater accuracy than the arithmetic mean temperature difference for systems exhibiting exponential temperature profiles, as the latter assumes uniform linear changes that overestimate the average driving force in most practical scenarios. For non-counterflow configurations, such as cross-flow exchangers, a correction factor F (typically between 0 and 1) is applied to the counterflow LMTD to account for geometric and flow arrangement effects, enabling reliable sizing via Q = U A F \Delta T_{lm}. This approach has been a staple in thermal engineering since the early 20th century, prominently featured in seminal texts like McAdams' Heat Transmission (first edition, 1933). An analogous logarithmic mean concentration difference applies in mass transfer processes, particularly for diffusion across stagnant films or in absorption/desorption systems, where the driving force is the logarithmic average of concentration gradients at the boundaries, mirroring the heat transfer derivation to quantify flux rates accurately.

Other Uses

In statistics and energy economics, the logarithmic mean is central to the Logarithmic Mean Divisia Index (LMDI), a decomposition technique used to attribute changes in aggregate indicators, such as energy consumption or emissions, to underlying factors like activity levels, structure, and intensity in time-series data. LMDI employs the logarithmic mean as a weighting factor to ensure perfect decomposition without residuals, making it preferable for multi-factor analyses in policy evaluation. For instance, the change in an indicator ΔV due to a factor x is given by ΔV_x = ∑ L(V_{i,t}, V_{i,t-1}) \ln(x_{i,t} / x_{i,t-1}), where L(a,b) = (a - b) / \ln(a / b) is the logarithmic mean and the sum is over categories i. In numerical analysis, the logarithmic mean provides a stable approximation for computing ratios in flux functions, particularly in high-order schemes for solving hyperbolic partial differential equations, such as those modeling compressible flows. It avoids numerical instabilities like division by near-zero values when left and right states are similar, as in entropy-stable discretizations where the logarithmic mean of densities or pressures ensures robust handling of discontinuities.

Relations to Other Means

Comparisons with Standard Means

The logarithmic mean L(x, y) of two positive real numbers x > y > 0 satisfies the inequality H(x, y) < L(x, y) < A(x, y), where H(x, y) = \frac{2xy}{x + y} is the and A(x, y) = \frac{x + y}{2} is the . This positions the logarithmic mean strictly above the harmonic mean, which provides a conservative lower estimate for rates or averages in certain contexts, but below the arithmetic mean, which tends to overestimate when values differ significantly. Additionally, the geometric mean G(x, y) = \sqrt{xy} serves as a lower bound for the logarithmic mean, with G(x, y) < L(x, y) < A(x, y). The logarithmic mean thus lies between the geometric and arithmetic means, offering a refinement of the classical AM-GM inequality by capturing an intermediate value that better approximates certain nonlinear averaging scenarios. This positioning highlights its role as a "logarithmic average" that bridges the multiplicative nature of the geometric mean and the additive nature of the arithmetic mean. In the limit as \left| \ln(x/y) \right| \to 0 (i.e., x/y \to 1), the logarithmic mean converges to the common value x = y, which coincides with the limit of the . For large ratios x/y \to \infty, asymptotic expansions show that L(x, y) grows slower than the but faster than the geometric mean, with behavior approximated by L(x, y) \sim x / \ln(x/y), adjusting for the logarithmic scale of divergence. The logarithmic mean is particularly suitable for averaging quantities that exhibit exponential variation, as its definition inherently incorporates the logarithm, providing a more accurate representation than linear means like the for such processes.

Connections to Advanced Means

The logarithmic mean fits within the broader hierarchy of power means, where the power mean of order p is defined as M_p(x, y) = \left( \frac{x^p + y^p}{2} \right)^{1/p} for p \neq 0, with the limit as p \to 0 yielding the geometric mean. The logarithmic mean is distinct from this family but lies strictly between the geometric mean (p = 0) and the arithmetic mean (p = 1), while the harmonic mean corresponds to p = -1 and the quadratic mean to p = 2. Specifically, for distinct positive x, y > 0, the ordering is M_{-1}(x, y) < M_0(x, y) < L(x, y) < M_1(x, y) < M_2(x, y). Inequalities involving the logarithmic mean extend to power means of higher orders, with M_p(x, y) > M_1(x, y) > L(x, y) for p > 1. The Stolarsky mean generalizes the logarithmic mean within a parametric family of means. Defined for p \neq 1 as S_p(x, y) = \left( \frac{x^p - y^p}{p (x - y)} \right)^{1/(p-1)}, the Stolarsky mean reduces to the logarithmic mean in the case p = 1, obtained via the limit L(x, y) = \lim_{p \to 1} S_p(x, y) = \frac{x - y}{\ln x - \ln y}. This parametrization positions the logarithmic mean as a specific instance in a broader class that interpolates between various symmetric means, including the as p \to 0. The logarithmic mean also appears in integral identities relating symmetric sums, such as the difference between the and geometric means. One such representation expresses A(x, y) - G(x, y) through an involving the logarithmic mean, highlighting its role in bridging classical means via continuous forms. For instance, A(x, y) - L(x, y) = \frac{x - y}{\pi} \int_0^\infty \frac{P_{x,y}(s)}{s} e^{-s y} \, ds, where P_{x,y}(s) encapsulates kernel functions tied to the mean's structure.