section_path: Logarithm tables, slide rules, and historical applications/Log tables
section_name: Log tables
paragraph_text: Log tables
  
  A key tool that enabled the practical use of logarithms was the table of logarithms. The first such table was compiled by Henry Briggs in 1617, immediately after Napier's invention but with the innovation of using 10 as the base. Briggs' first table contained the common logarithms of all integers in the range from 1 to 1000, with a precision of 14 digits. Subsequently, tables with increasing scope were written. These tables listed the values of log x for any number x in a certain range, at a certain precision. Base-10 logarithms were universally used for computation, hence the name common logarithm, since numbers that differ by factors of 10 have logarithms that differ by integers. The common logarithm of x can be separated into an integer part and a fractional part, known as the characteristic and mantissa. Tables of logarithms need only include the mantissa, as the characteristic can be easily determined by counting digits from the decimal point. The characteristic of 10 · x is one plus the characteristic of x, and their mantissas are the same. Thus using a three-digit log table, the logarithm of 3542 is approximated by
  
  log 10 3542 = log 10 (1000 ⋅ 3.542) = 3 + log 10 3.542 ≈ 3 + log 10 3.54 {\displaystyle \log _{10}3542=\log _{10}(1000\cdot 3.542)=3+\log _{10}3.542\approx 3+\log _{10}3.54}
  
  Greater accuracy can be obtained by interpolation:
  
  log 10 3542 ≈ 3 + log 10 3.54 + 0.2 (log 10 3.55 − log 10 3.54) {\displaystyle \log _{10}3542\approx 3+\log _{10}3.54+0.2(\log _{10}3.55-\log _{10}3.54)}
  
  The value of 10 can be determined by reverse look up in the same table, since the logarithm is a monotonic function.