Editing Logarithm (section)

===Computations===
The product and quotient of two positive numbers {{Mvar|c}} and ''{{Mvar|d}}'' were routinely calculated as the sum and difference of their logarithms. The product&nbsp;{{Math|''cd''}} or quotient&nbsp;{{Math|''c''/''d''}} came from looking up the antilogarithm of the sum or difference, via the same table:

<math display="block"> cd = 10^{\, \log_{10} c} \, 10^{\,\log_{10} d} = 10^{\,\log_{10} c \, + \, \log_{10} d}</math>
and
<math display="block">\frac c d = c d^{-1} = 10^{\, \log_{10}c \, - \, \log_{10}d}.</math>

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 [[nth root|roots]] are reduced to multiplications or divisions and lookups by
<math display="block">c^d = \left(10^{\, \log_{10} c}\right)^d =  10^{\, d \log_{10} c}</math>

and
<math display="block">\sqrt[d]{c} = c^\frac{1}{d} = 10^{\frac{1}{d} \log_{10} c}.</math>

Trigonometric calculations were facilitated by tables that contained the common logarithms of [[trigonometric function]]s.