The term ''new'' is not appropriate, I just noted the existence of a similar formula at http://functions.wolfram.com/ElementaryFunctions/Log/09/, where we find : ln(x) = Limit(w*(x^(1/w)-1),n=infinity) ; The formula 35 that I proposed could therefore be an improvement of the formula above, in fact: With the formula above: r:=evalf (2^(40)*(2^(1/2^(40))-1),40); r := .69314718056016379415253688694241624546042537621 With the proposed improvement, for x = 2 and n = 40: r1:=evalf (2^(40)*(2^(1/2^(40))-1)/(2^(1/2^(40+1))),40); r1 := .6931471805599453094172321334848075670387 By comparing: Ln(2)-r := -.2184847353047654842396773849e-12 Ln(2)-r1 := -.120266309989632e-25 Which in this case seems to indicate a double gain of decimals obtained.