Chapter 8 - Exponential and Logarithmic Functions - Exponential and Logarithmic Functions, Rules of Logarithms and Exponential Growth

Oct 16, 2026

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Chapter 8 explains exponential functions by treating x as an exponent, starting with f(x)=2^x and building a table of values to show rapid growth as x increases. It derives exponent rules for multiplying exponential expressions, including the addition property xa·xb=xa+b and the power-of-a-power (x^a)^b=x^(ab). The chapter extends these ideas to negative exponents and the use of the denominator rule, then discusses x=0 and fractional exponents as roots, leading to continuous graph behavior and the asymptote at the x-axis.

LanguageEnglish

CategoryExam

FormatPDF

Pages29

Reading Time1 h 13 min

AccessFree to download

Exponential Functions

Defining f(x)=2^x and value tables

Zero and negative exponents

Rules of exponents: addition and power-of-a-power

Negative exponent definition and reciprocal form

Fractional exponents and roots

Graph behavior and asymptotes

Exponential Function Transformations

Shifting inputs and rewriting with exponent rules

Chapter 8 Exponential and Logarithmic Functions, Rules of Logarithms and Exponential Growth Exponential Functions Now that we know what inverse functions are, it’s time to apply them to some very interesting and useful functions called exponential functions. Exponential functions are called this because the variable x is or is a part of the exponent of a number. A good starting example of this is f (x) = 2x . Here, I am always taking the number 2 and multiplying it a certain number of times. Let’s build a table to help us see what the graph will look like: | x | f (x) = 2x | | --- | --- | | 0 | 1 | | 1 | 2 | | 2 | 4 | | 3 | 8 | Notice that with each x-value, I’m taking the number 2 to that power. 2 to the 0 power is 1. That’s because anything to the zero power is 1. I will show you why in a bit. 2 to the 1st power is 2. 2 to the 2nd power is 4. 22 = 2 × 2 = 4. 2 to the 3rd power is 8. 23 = 2 × 2 × 2 = 8. If we continue, when x is 4, y is 16; when x is 5, y is 32, and so on. You can see that as x gets bigger, the y-values get really bigger, and in a hurry. Before we go on, I want to make sure you understand what happens when we have 0 as an exponent, or a negative number as an exponent. To do this well, I want to go all the way back to how exponents work in the first place. Let’s suppose that we are multiplying x 2 × x 3 . We can rewrite this as [x × x][x × x × x] , which can be re-written as x × x × x × x × x , which equals x 5 . Thus, to multiply exponential expressions like this, we see that we add the exponents. To say this more generally, xa xb = xa+b We call this the Addition Property of Exponents. Now watch what happens when we put the exponential expression within parentheses: (x2)3 = (x2)(x2)(x2) = (x × x)(x × x)(x × x) = x 6 Here, the exponents get multiplied, but only because one of them is contained within parentheses. Our general rule, then, is (xa )b = xab. Let’s put these two rules together to see what we get with and have 22 23 . We re-write this 2~~ ~~~~ ~~×2~~ ~~~~ ~~2 , which we can cancel two pairs of twos to get ~~ ~~12 . Thus, if we define negative exponents to mean that the number with the negative exponent goes in the denominator of the expression, with a positive exponent, then we’ve got something. In the above example, 22 23 = 22-3 = 2-1 . We thus define negative exponents to work like this: x-a = ~~ ~~x1a . Using our second rule, the one with the parentheses, we would then have x-(-a) = xa . At this point I feel that I should apologize for the difficulties in writing this chapter. The various fonts available do not permit me to write exponents as neatly as I would like! We get the same result for the above if we do the negative exponent rule alone, and write it as 1 1 xa a = x . Finally, what about when x = 0 is our exponent? Again, using the addition rule of exponents, we could ask ourselves what is ~~ ~~2525 ? 1 = 2525 = 25-5 = 20 . Thus we can add more points to our table: | x | f (x) = 2x | | --- | --- | | 0 | 1 | | 1 | 2 | | 2 | 4 | | 3 | 8 | | -1 | 1/2 | | -2 | 1/4 | | -3 | 1/8 | | -4 | 1/16 | Notice something about these values: even though the x-values can be zero or negative, the y-values are always positive. If we graph these points, we get the following: So far we have a series of dots that are based on the integers we have fed into our function f (x) = 2x . Are there more points besides these? We can put in quite a few more, and here is one way to look at it. Suppose I let 1 1 x = 2 . Then what would 2 2 be equal to (please read that as “two to the one-half power”, not “two and a half”)? Let’s reason it out again by using our addition property of exponents. Let’s multiply 212 times itself and see what we get: 212 × 212 = 2 1/2+1/2 = 21 = 2 . The only number that, times itself, is equal to 2 is 2 : 2 × 2 = 2 by definition of the square root. Thus we determine that fractions in our

FAQs

How does the chapter define exponential functions using exponents?

It defines exponential functions as those in which the variable x is in the exponent, using the example f(x)=2^x to generate values by repeatedly multiplying the base.

What is the addition property of exponents and where is it used?

The addition property states xa·xb=xa+b. The chapter uses it to simplify products of exponential expressions like x^2·x^3.

How does the chapter explain negative and fractional exponents?

Negative exponents are defined so that the expression with a negative exponent goes into the denominator, effectively forming a reciprocal. Fractional exponents are treated as roots, justified by reasoning about square roots and related powers.

Oct 16, 2026

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