In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer.

Definition

The logarithmic mean is defined by

.

Inequalities

The logarithmic mean of two numbers is smaller than the arithmetic mean and the generalized mean with exponent greater than 1. However, it is larger than the geometric mean and the harmonic mean, respectively. The inequalities are strict unless both numbers are equal..

Derivation

Mean value theorem of differential calculus

From the mean value theorem, there exists a value in the interval between and where the derivative equals the slope of the secant line

The logarithmic mean is obtained as the value of by substituting for and similarly for its corresponding derivative

and solving for

Integration

This interpretation allows the derivation of some properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by and .

Generalization

Mean value theorem of differential calculus

One can generalize the mean to variables by considering the mean value theorem for divided differences for the -th derivative of the logarithm.

denotes a divided difference of the logarithm. For

Integral

The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex

In the case of

Connection to other means

Some other means can be expressed in terms of the logarithmic mean.

See also

• The logarithmic mean is a special case of the Stolarsky mean.

• Logarithmic mean temperature difference

• Log semiring

References

1. Citations

   Bibliography

2. Oilfield Glossary: Term 'logarithmic mean'