Natural logarithms

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Logarithms with base e, where e is Euler's number, are called natural logarithms. We denote the natural logarithm of X by LN(X).

ALGORITHM

For 0<X<1 is LN(X)=-LN(1/X)
For X>1 find X=(2**M)*Z
Then

LN(X)=M*LN(2)-2*(Zeta/1+Zeta**3/3+Zeta**5/5+...)

IMPLEMENTATION

Unit: internal function, external function without procedure statement
 
Parameters: a real number X>0, a positive integer P - number of significant digits of LN(X), default is 9
 
Interface: LN2 function. For P<=200 we can use instead LN2Ponly the LN2 function
 
Returns: the first P significant decimal digits of LN(X)

 

LN: procedure parse arg X, P if P = "" then P = 9; numeric digits P if X < 1 then return - LN(1 / X, P) do M = 0 until (2 ** M) > X; end M = M - 1 Z = X / (2 ** M) Zeta = (1 - Z) / (1 + Z) N = Zeta; Ln = Zeta; Zetasup2 = Zeta * Zeta do J = 1   N = N * Zetasup2; NewLn = Ln + N / (2 * J + 1)   if NewLn = Ln then return M * LN2P(P) - 2 * Ln   Ln = NewLn end

 

 

LN2P: procedure parse arg P if P <= 200 then return LN2() N = 1 / 3; Ln = N; Zetasup2 = 1 / 9 do J = 1   N = N * Zetasup2; NewLn = Ln + N / (2 * J + 1)   if NewLn = Ln then return 2 * Ln   Ln = NewLn end

 

CONNECTIONS

Euler's number e
Euler's constant gamma
Logarithms with other bases than e
Technique: Beforehand computed constants
P+n trick by Walter Pachl in Reflexio

Literature

Jarník V. Diferenciální počet I

Nakladatelství České Akademie Věd, Praha, 1963

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