Double exponential function

A double exponential function is a constant raised to the power of an exponential function. The general formula is ${\displaystyle f(x)=a^{b^{x}}=a^{(b^{x})}}$ (where a>1 and b>1), which grows much more quickly than an exponential function. For example, if a = b = 10:

• f(x) = 10^10x
• f(0) = 10
• f(1) = 10¹⁰
• f(2) = 10¹⁰⁰ = googol
• f(3) = 10¹⁰⁰⁰
• f(100) = 10¹⁰¹⁰⁰ = googolplex.

Factorials grow faster than exponential functions, but much more slowly than doubly exponential functions. However, tetration and the Ackermann function grow faster. See Big O notation for a comparison of the rate of growth of various functions.

The inverse of the double exponential function is the double logarithm log(log(x)).