section_path: Analytic properties/Characterization by the product formula
section_name: Characterization by the product formula
paragraph_text: Characterization by the product formula
  
  The function log x can also be essentially characterized by the product formula log b (x y) = log b x + log b y. {\displaystyle \log _{b}(xy)=\log _{b}x+\log _{b}y.} More precisely, the logarithm to any base b > 1 is the only increasing function f from the positive reals to the reals satisfying f (b) = 1 and f (x y) = f (x) + f (y). {\displaystyle f(xy)=f(x)+f(y).}