section_path: Logarithmic identities/Change of base
section_name: Change of base
paragraph_text: Change of base
  
  The logarithm log x can be computed from the logarithms of x and b with respect to an arbitrary base k using the following formula: log b x = log k x log k b. {\displaystyle \log _{b}x={\frac {\log _{k}x}{\log _{k}b}}.}
  
  Typical scientific calculators calculate the logarithms to bases 10 and e. Logarithms with respect to any base b can be determined using either of these two logarithms by the previous formula: log b x = log 10 x log 10 b = log e x log e b. {\displaystyle \log _{b}x={\frac {\log _{10}x}{\log _{10}b}}={\frac {\log _{e}x}{\log _{e}b}}.}
  
  Given a number x and its logarithm y = log x to an unknown base b, the base is given by:
  
  b = x 1 y, {\displaystyle b=x^{\frac {1}{y}},}
  
  which can be seen from taking the defining equation x = b log b x = b y {\displaystyle x=b^{\,\log _{b}x}=b^{y}} to the power of 1 y. {\displaystyle {\tfrac {1}{y}}.}