Logarithm \Log"a*rithm\ (l[o^]g"[.a]*r[i^][th]'m), n. [Gr.
   lo`gos word, account, proportion + 'ariqmo`s number: cf. F.
   logarithme.] (Math.)
   One of a class of auxiliary numbers, devised by John Napier,
   of Merchiston, Scotland (1550-1617), to abridge arithmetical
   calculations, by the use of addition and subtraction in place
   of multiplication and division.

   Note: The relation of logarithms to common numbers is that of
         numbers in an arithmetical series to corresponding
         numbers in a geometrical series, so that sums and
         differences of the former indicate respectively
         products and quotients of the latter; thus,
         0 1 2 3 4 Indices or logarithms
         1 10 100 1000 10,000 Numbers in geometrical progression
         Hence, the logarithm of any given number is the
         exponent of a power to which another given invariable
         number, called the base, must be raised in order to
         produce that given number. Thus, let 10 be the base,
         then 2 is the logarithm of 100, because 10^{2} = 100,
         and 3 is the logarithm of 1,000, because 10^{3} =
         1,000.
         
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   {Arithmetical complement of a logarithm}, the difference
      between a logarithm and the number ten.

   {Binary logarithms}. See under {Binary}.

   {Common logarithms}, or {Briggs logarithms}, logarithms of
      which the base is 10; -- so called from Henry Briggs, who
      invented them.

   {Gausss logarithms}, tables of logarithms constructed for
      facilitating the operation of finding the logarithm of the
      sum of difference of two quantities from the logarithms of
      the quantities, one entry of those tables and two
      additions or subtractions answering the purpose of three
      entries of the common tables and one addition or
      subtraction. They were suggested by the celebrated German
      mathematician Karl Friedrich Gauss (died in 1855), and are
      of great service in many astronomical computations.

   {Hyperbolic logarithm} or {Napierian logarithm} or {Natural  logarithm}, a logarithm (devised by John Speidell, 1619) of
      which the base is e (2.718281828459045...); -- so called
      from Napier, the inventor of logarithms.

   {Logistic logarithms} or {Proportional logarithms}, See under
      {Logistic}.
      
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 Logarithmetic

Binary \Bi"na*ry\, a. [L. binarius, fr. bini two by two, two at
   a time, fr. root of bis twice; akin to E. two: cf. F.
   binaire.]
   Compounded or consisting of two things or parts;
   characterized by two (things).
   
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   {Binary arithmetic}, that in which numbers are expressed
      according to the binary scale, or in which two figures
      only, 0 and 1, are used, in lieu of ten; the cipher
      multiplying everything by two, as in common arithmetic by
      ten. Thus, 1 is one; 10 is two; 11 is three; 100 is four,
      etc. --Davies & Peck.

   {Binary compound} (Chem.), a compound of two elements, or of
      an element and a compound performing the function of an
      element, or of two compounds performing the function of
      elements.

   {Binary logarithms}, a system of logarithms devised by Euler
      for facilitating musical calculations, in which 1 is the
      logarithm of 2, instead of 10, as in the common
      logarithms, and the modulus 1.442695 instead of .43429448.
      

   {Binary measure} (Mus.), measure divisible by two or four;
      common time.

   {Binary nomenclature} (Nat. Hist.), nomenclature in which the
      names designate both genus and species.

   {Binary scale} (Arith.), a uniform scale of notation whose
      ratio is two.

   {Binary star} (Astron.), a double star whose members have a
      revolution round their common center of gravity.

   {Binary theory} (Chem.), the theory that all chemical
      compounds consist of two constituents of opposite and
      unlike qualities.
      
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