## 5. Exponential and logarithmic functions for any positive real base.

From (4.8) *ln a*^{x} =x ln a, we have

Exponential functions *a*^{x}, with base *a* other than *e*,
equal a natural exponential function in which the exponent *x* is multiplied by the
natural logarithm of the base *a*.

However, since *ln a* is a real function only if *a>0*, such functions are
real only if *a>0*. Since *1*^{x}=1 for whatever *x*,
if *a*=1 the exponential cannot be inverted. Otherwise, these functions inherit the
same properties of the natural exponential: they are always positive and monotonic
(increasing if *a*>1, decreasing if *a*<1), so they are invertible.

The following Javascript application allows you to tabulate and graph exponential functions with real base.

The natural base can be entered in the input fields as *E* (uppercase).

If your browser does not allow internal frames, you can directly access the application page.

The browser must allow pop-up windows.

The inverse functions of each of these exponential functions is said
**logarithm to base ***a* and is denoted by
*log*_{a}.

The logarithm to base 10, said common logarithm, are often denoted by
*Log*.

By definition

From the second one of the (5.2) we have

The logarithm to base *a* of a real number *x* is directly proportional to
the natural logarithm of *x*; it equals the ratio between this natural logarithm and
the logarithm of *a*.

Since the logarithm to base *a* is proportional to natural logarithm,
it inherits all its algebraic properties. For example:

that is

whatever the base, the logarithm of a product equals the sum of the logarithms of its
factors.

The following Javascript application allows you to tabulate and graph logarithmic functions with real base.

The natural base can be entered in the input fields as *E* (uppercase).

If your browser does not allow internal frames, you can directly access the application page.

The browser must allow pop-up windows.