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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. In symbols:
Contents
- Inequalities
- Mean value theorem of differential calculus
- Integration
- Integral
- Connection to other means
- References
for the positive numbers
Inequalities
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):
Mean value theorem of differential calculus
From the mean value theorem
the logarithmic mean is obtained as the value of
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
Mean value theorem of differential calculus
You can generalize the mean to
where
For
Integral
The integral interpretation can also be generalized to more variables, but it leads to a different result. Given the simplex
This can be simplified using divided differences of the exponential function to
Example