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Revision as of 15:42, 21 June 2020 by Misterbill999(talk | contribs)(Added "the slope of" before "the secant line" in the section headed "Mean value theorem of differential calculus.")
The logarithmic mean of two numbers is smaller than the arithmetic mean but larger than the geometric mean (unless the numbers are the same, in which case all three means are equal to the numbers):
The logarithmic mean is obtained as the value of by substituting for and similarly for its corresponding derivative:
and solving for :
Integration
The logarithmic mean can also be interpreted as the area under an exponential curve.
The area interpretation allows the easy derivation of some basic properties of the logarithmic mean. Since the exponential function is monotonic, the integral over an interval of length 1 is bounded by and . The homogeneity of the integral operator is transferred to the mean operator, that is .
The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex with and an appropriate measure which assigns the simplex a volume of 1, we obtain
This can be simplified using divided differences of the exponential function to
^
B. Ostle; H. L. Terwilliger (1957). "A comparison of two means". Proc. Montana Acad. Sci. 17: 69–70. {{cite journal}}: Unknown parameter |last-author-amp= ignored (|name-list-style= suggested) (help)