## Quick Reference

The logarithmic function ln must be distinguished from the logarithmic function to base *a* (see logarithm). Here are two approaches:

1.

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 log_{e}*x*, and the logarithmic function ln can be taken to be just this function log_{e}. 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.

2.

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:*xy*)=ln *x*+ln *y*.*x*) =−ln *x*.*x*/*y*)=ln *x*−ln *y*.*x*^{r})=*r* ln *x*.With this approach, exp can be defined as the inverse function of ln, and the number *e* defined as exp 1. Finally, it is shown that ln *x* and log_{e}*x* are identical.

*xy*)=ln *x*+ln *y*.

*x*) =−ln *x*.

*x*/*y*)=ln *x*−ln *y*.

*x*^{r})=*r* ln *x*.

*Subjects:*
Mathematics.

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