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 double 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)). The complex double exponential function is entire, due to the fact that it is the composition of two entire functions ${\displaystyle f(x)=a^{x}=e^{x\ln a}}$ and ${\displaystyle g(x)=b^{x}=e^{x\ln b}}$.