The geometric mean is a type of power mean. For a collection of positive real numbers, their geometric mean is defined to be
For instance, the geometric mean of and is
and the geometric mean of , , and is
The terminology geometric mean comes from the fact that this quantity has a simple geometric interpretation. Consider the below picture, where and .
From the Pythagorean theorem, one obtains the three equations
and substituting , gives
In particular, consider drawing a semicircle with diameter that encapsulates the given figure. Then the radius of the semicircle is , and the radius must be greater than or equal to (where equality occurs only when is directly above the semicircle's center). Hence,
for all positive real numbers and . This is the two-variable case of the AM-GM inequality.
Observe that by taking logarithms, we have
Hence, the logarithm of the geometric mean is the arithmetic mean of the logarithms.