section_path: Logarithm tables, slide rules, and historical applications/Computations
section_name: Computations
paragraph_text: Computations
  
  The product and quotient of two positive numbers c and d were routinely calculated as the sum and difference of their logarithms. The product cd or quotient c / d came from looking up the antilogarithm of the sum or difference, via the same table:
  
  c d = 10 log 10 c 10 log 10 d = 10 log 10 c + log 10 d {\displaystyle cd=10^{\,\log _{10}c}\,10^{\,\log _{10}d}=10^{\,\log _{10}c\,+\,\log _{10}d}} and c d = c d − 1 = 10 log 10 c − log 10 d. {\displaystyle {\frac {c}{d}}=cd^{-1}=10^{\,\log _{10}c\,-\,\log _{10}d}.}
  
  For manual calculations that demand any appreciable precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier methods such as prosthaphaeresis, which relies on trigonometric identities.
  
  Calculations of powers and roots are reduced to multiplications or divisions and lookups by c d = (10 log 10 c) d = 10 d log 10 c {\displaystyle c^{d}=\left(10^{\,\log _{10}c}\right)^{d}=10^{\,d\log _{10}c}}
  
  and c d = c 1 d = 10 1 d log 10 c. {\displaystyle {\sqrt{c}}=c^{\frac {1}{d}}=10^{{\frac {1}{d}}\log _{10}c}.}
  
  Trigonometric calculations were facilitated by tables that contained the common logarithms of trigonometric functions.