Double exponential function

A double exponential function is a <a href="/facts/Constant_(mathematics)/SxlxTWFT">constant</a> raised to the power of an <a href="/facts/Exponentiation/pac6QY2g">exponential function</a>. The general formula is 
 
 
 
 f
 (
 x
 )
 =
 
 a
 
 
 b
 
 x
 
 
 
 
 =
 
 a
 
 (
 
 b
 
 x
 
 
 )
 
 
 
 
 {\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:

<ul><li>f(x) = 1010x</li>
<li>f(0) = 10</li>
<li>f(1) = 1010</li>
<li>f(2) = 10100 = <a href="/facts/Googol/FPxsLCU8">googol</a></li>
<li>f(3) = 101000</li>
<li>f(100) = 1010100 = <a href="/facts/Googolplex/MoGnKqMH">googolplex</a>.</li></ul>
<a href="/facts/Factorial/kkfy1Bwe">Factorials</a> grow faster than exponential functions, but much more slowly than double exponential functions. However, <a href="/facts/Tetration/5rKeVVYv">tetration</a> and the <a href="/facts/Ackermann_function/WQNpCWHn">Ackermann function</a> grow faster. See <a href="/facts/Big_O_notation/weFFjSWg">Big O notation</a> for a comparison of the rate of growth of various functions.
The inverse of the double exponential function is the <a href="/facts/Logarithm/QeXaV97q">double logarithm</a> log(log(x)). The complex double exponential function is <a href="/facts/Entire_function/XNluKzu7">entire</a>, because it is the composition of two entire functions 
 
 
 
 f
 (
 x
 )
 =
 
 a
 
 x
 
 
 =
 
 e
 
 x
 ln
 ⁡
 a
 
 
 
 
 {\displaystyle f(x)=a^{x}=e^{x\ln a}}
 
 and 
 
 
 
 g
 (
 x
 )
 =
 
 b
 
 x
 
 
 =
 
 e
 
 x
 ln
 ⁡
 b
 
 
 
 
 {\displaystyle g(x)=b^{x}=e^{x\ln b}}
 
.

Double exponential function open-in-new

Double exponential function