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Logarithmic and Exponential Functions


When logarithms were introduced in the seventeenth century as a computational tool, they provided scientists at that time with unimaginable calculating power.

Although computers and calculators have largely replaced logarithms in numerical calculations, logarithmic and relative functions have wide application in mathematics and science.

Irrational Exponents

In algebra, the integer and rational powers of a number B are defined by

If B negative, then some of the fractional powers of B will have imaginary values; for example, . To avoid this complication, let's assume that even if not explicitly stated.

Note that the preceding definitions do not include powers unreasonable in B, such as

There are several methods for defining irrational powers. One approach is to define irrational powers of B as a limit of rational powers. For example, to define we should start with the decimal representation of , this is,

3,1415926

From this decimal we can form a sequence of rational numbers that get closer and closer to this is,

3,1; 3,14; 3,141; 3,1415; 3,14159

and from these we can form a power sequence rational of 2:

Since the exponents of the terms of this sequence tend to a limit , it seems plausible that the terms themselves tend to a limit; therefore it is reasonable define as this limit. The table below provides numerical evidence that the sequence actually has a limit and for four decimal places the value of this limit is 8.8250. In general, for any irrational exponent P and positive number Bwe can define as the limit of rational powers of B, created by the decimal expansion of P.

Table

x
38,000000
3,18,574188
3,148,815241
3,1418,821353
3,14158,824411
3,141598,824962
3,1415928,824974

The family of exponential functions

A function of form f (x) = , Where B > 0 and b 1 is called base exponential function b, whose examples are

f (x) = , f (x) = , f (x) =

Note that an exponential function has a constant base and a variable exponent. Thus functions such as f (x) = and f (x) = would not be classified as exponential functions since they have a variable base and a constant exponent.

It can be shown that exponential functions are continuous and have one of the two basic aspects shown in figure 1depending on whether 0 < B <1 or B > 1. Figure 2 shows graphs of some specific exponential functions.

NOTE. If B = 1, then the function is constant since = = 1. This case is not in our interest here, so we exclude it from the family of exponential functions.

Next: Logarithms