 He discovered ways to express various logarithmic functions in terms of power series, and successfully defined logarithms for complex and negative numbers, thus greatly expanding the scope where logarithms could be applied in mathematics.Most researchers in the field long held the view that formula_7 for any positive real formula_8 since by using the additivity property of logarithms formula_9.In a 1747 letter to Jean Le Rond d'Alembert, Euler defined the natural logarithm of −1 as formula_10 a pure imaginary.