section_path: Analytic properties/Existence
section_name: Existence
paragraph_text: Existence
  
  Let b be a positive real number not equal to 1 and let f (x) = b.
  
  It is a standard result in real analysis that any continuous strictly monotonic function is bijective between its domain and range. This fact follows from the intermediate value theorem. Now, f is strictly increasing (for b > 1), or strictly decreasing (for 0 < b < 1), is continuous, has domain R {\displaystyle \mathbb {R} }, and has range R > 0 {\displaystyle \mathbb {R} _{>0}}. Therefore, f is a bijection from R {\displaystyle \mathbb {R} } to R > 0 {\displaystyle \mathbb {R} _{>0}}. In other words, for each positive real number y, there is exactly one real number x such that b x = y {\displaystyle b^{x}=y}.
  
  We let log b: R > 0 → R {\displaystyle \log _{b}\colon \mathbb {R} _{>0}\to \mathbb {R} } denote the inverse of f. That is, log y is the unique real number x such that b x = y {\displaystyle b^{x}=y}. This function is called the base- b logarithm function or logarithmic function (or just logarithm).