Gaussian logarithm

In mathematics, addition and subtraction logarithms or Gaussian logarithms can be utilized to find the logarithms of the sum and difference of a pair of values whose logarithms are known, without knowing the values themselves.[1]

Their mathematical foundations trace back to Zecchini Leonelli[2][3] and Carl Friedrich Gauss[4][1][5] in the early 1800s.[2][3][4][1][5]

The ${\displaystyle s_{b}(z)}$ and ${\displaystyle d_{b}(z)}$ functions for ${\displaystyle b=e}$.

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

${\displaystyle \log _{b}(|X|+|Y|)=x+s_{b}(y-x)}$

${\displaystyle \log _{b}(||X|-|Y||)=x+d_{b}(y-x),}$

where ${\displaystyle x=\log _{b}(X)}$, ${\displaystyle y=\log _{b}(Y)}$, the "sum" function is defined by ${\displaystyle s_{b}(z)=\log _{b}(1+b^{z})}$, and the "difference" function by ${\displaystyle d_{b}(z)=\log _{b}(|1-b^{z}|)}$. The functions ${\displaystyle s_{b}(z)}$ and ${\displaystyle d_{b}(z)}$ are also known as Gaussian logarithms.

For natural logarithms with ${\displaystyle b=e}$ the following identities with hyperbolic functions exist:

${\displaystyle s_{e}(z)=\ln 2+{\frac {z}{2}}+\ln \left(\cosh {\frac {z}{2}}\right)}$

${\displaystyle d_{e}(z)=\ln 2+{\frac {z}{2}}+\ln \left|\sinh {\frac {z}{2}}\right|}$

This shows that ${\displaystyle s_{e}}$ has a Taylor expansion 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.