It is also the unique positive number a such that the graph of the function y = a x has a slope of 1 at x = 0 .
The (natural) exponential function f ( x ) = e x is the unique function f that equals its own derivative and satisfies the equation f (0) = 1 ; hence one can also define e as f (1) . The natural logarithm, or logarithm to base e , is the inverse function to the natural exponential function. The natural logarithm of a number k > 1 can be defined directly as the area under the curve y = 1/ x between x = 1 and x = k , in which case e is the value of k for which this area equals 1 (see image). There are various
The first known use of the constant, [ clarification needed ] represented by the letter b , was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. [10]