The logarithmic function ln must be distinguished from the logarithmic function to base a (see logarithm). Here are two approaches:
Suppose that the value of the number e has already been obtained independently. Then the logarithm of x to base e can be defined and denoted by logex, and the logarithmic function ln can be taken to be just this function loge. The problem with this approach is its reliance on a prior definition of e and the difficulty of subsequently proving some of the important properties of ln.
The following is more satisfactory. Let f be the function defined, for t>0, by f(t)=1/t. Then the logarithmic function ln is defined, for x>0, by
The intention is best appreciated in the case when x>1, for then ln x gives the area under the graph of f in the interval [1, x]. This function ln is continuous and increasing; it is differentiable and, from the fundamental relationship between differentiation and integration, its derived function is the function f. Thus it has been established that From the definition, the following properties can be obtained, where x, y and r are real, with x>0 and y>0: