page_title: E (mathematical constant)
section_title: Exponential-like functions
hierarchical_section_title: E (mathematical constant) / Properties / Exponential-like functions
context_page_description: The number e, known as Euler's number, is a mathematical constant approximately equal to 2.71828, and can be characterized in many ways. It is the base of the natural logarithm. It is the limit of ⁿ as n approaches infinity, an expression that arises in the study of compound interest. It can also be calculated as the sum of the infinite series
  It is also the unique positive number a such that the graph of the function y = aˣ has unit slope at x = 0.
  The exponential function f(x) = eˣ is the unique function which is equal to its own derivative, with the initial value f = 1. The natural logarithm, or logarithm to base e, is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/x between x = 1 and x = k, in which case e is the value of k for which this area equals one. There are various other characterizations.
  e is sometimes called Euler's number, after the Swiss mathematician Leonhard Euler, or as Napier's constant. However, Euler's choice of the symbol e is said to have been retained in his honor.
context_section_description: Steiner's problem asks to find the global maximum for the function
  This maximum occurs precisely at x = e.
  The value of this maximum is 1.4446 6786 1009 7661 3365... (accurate to 20 decimal places).
  For proof, the inequality , from above, evaluated at  and simplifying gives .  So  for all positive x.
  Similarly, x = 1/e is where the global minimum occurs for the function
  defined for positive x.  More generally, for the function
  the global maximum for positive x occurs at x = 1/e for any n < 0; and the global minimum occurs at x = e-1/ⁿ for any n > 0.
  The infinite tetration
   or 
  converges if and only if e⁻ᵉ ≤ x ≤ e1/ᵉ (or approximately between 0.0660 and 1.4447), due to a theorem of Leonhard Euler.
caption: The global maximum of 
    
      
        
          
            
              x
              
                x
              
            
          
        
      
      {\displaystyle {\sqrt[{x}]{x}}}
    
   occurs at x = e.