What is the difference between geometric mean and arithmetic mean?

Question

Arithmetic mean of two numbers A and B is:

(A + B)/2

Geometric mean of two numbers A and B is:

sqrt(A*B)

The geometric mean can be rewritten as follows:

sqrt(A*B) = Exp{Log[sqrt(A*B)]} =

Exp{1/2 [Log(A) + Log(B)]} =

Exp[Arithmetic mean of logarithms].