In mathematics, addition and subtraction logarithms or Gaussian logarithms can be utilized to find the

logarithm In mathematics, the logarithm is the inverse function to exponentiation. That means the logarithm of a number to the base is the exponent to which must be raised, to produce . For example, since , the ''logarithm base'' 10 o ...

s of the sum and difference of a pair of values whose logarithms are known, without knowing the values themselves. Their mathematical foundations trace back to Zecchini Leonelli and

Carl Friedrich Gauss Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...

in the early 1800s. Gaussian logarithm

The operations of addition and subtraction can be calculated by the formula: :

\log_b(, X, +, Y, )=x+s_b(y-x)

\log_b(, , X, -, Y, , )=x+d_b(y-x),

where

x=\log_b(X)

y=\log_b(Y)

, the "sum" function is defined by

s_b(z)=\log_b(1+b^z)

, and the "difference" function by

d_b(z)=\log_b(, 1-b^z, )

. The functions

s_b(z)

and

d_b(z)

are also known as ''Gaussian logarithms''. For

natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...

s with

b = e

the following identities with

hyperbolic function In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the ...

s exist: :

s_e(z) = \ln 2 + \frac + \ln \left(\cosh\frac\right)

d_e(z) = \ln 2 + \frac + \ln \left, \sinh\frac\

This shows that

s_e

has a

Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...

where all but the first term are rational and all odd terms except the linear one are zero. The simplification of multiplication, division, roots, and powers is counterbalanced by the cost of evaluating these functions for addition and subtraction.

See also

References

Further reading