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David Terr
Ph.D. Math, UC Berkeley

>> 3.2. Logarithmic Functions

3.1. Exponential Functions

Thus far, the only functions we have discussed are polynomial and rational functions. In this chapter, we will discuss two more important classes of functions, namely exponential and logarithmic functions. We will start with the first of these.

An exponential function is a function of the form f(x) = bx, where b is an arbitrary positive real number, known as the base, and the argument x is known as the exponent. We have already encountered exponents in Chapter P. In Section P.2, we defined integer exponents of an arbitrary nonzero real base and in Section P.4 we extended our definition to cover arbitrary rational exponents of positive real bases. In fact, our definition may be extended to arbitrary real exponents of positive bases. The formal definition of such an extension requires the use of calculus, but for now it will suffice to note that this extension works because exponential functions are monotonic (either strictly increasing, strictly decreasing, or constant in the case of b=1) and that the rational numbers form a dense subset of the real numbers, that is, for any real number x, we can find a rational number which is arbitrarily close to x.

Consider the function f(x) = 2x, Using our knowledge of rational exponents, we can calculate f(x) for x increasing in intervals of 0.2. The following table lists the values of f(x) for x ranging from -3.0 to 3.2. Values of f(x) are rounded to the nearest hundredth.

x 2x   x 2x
-3.0
0.12

0.2
1.15
-2.8
0.14

0.4
1.32
-2.6
0.16

0.6
1.52
-2.4
0.19

0.8
1.74
-2.2
0.22

1.0
2.00
-2.0
0.25

1.2
2.30
-1.8
0.29

1.4
2.64
-1.6
0.33

1.6
3.03
-1.4
0.38

1.8
3.48
-1.2
0.44

2.0
4.00
-1.0
0.50

2.2
4.59
-0.8
0.57

2.4
5.28
-0.6
0.66

2.6
6.06
-0.4
0.76

2.8
6.96
-0.2
0.87

3.0
8.00
0.0
1.00

3.2
9.19

Below we plot a graph of this function.

Figure 3.1.1: Graph of f(x) = 2x

In Section P.4, we listed some useful properties of exponents. These properties apply for exponential functions as well. We list them again below, but with changes to the names of the variables and constants in order to follow our current discussion more closely:

• (3.1.1): bx+y = bxby
• (3.1.2): b-x = 1/bx
• (3.1.3): bx-y = bx/by
• (3.1.4): (bx)y = bxy
• (3.1.5): bxcx = (bc)x

>> 3.2. Logarithmic Functions