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12.2: Exponential functions

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Exponential functions take on their own set of solving and simplifying techniques since the equations are a bit different than before. For example, before we had something like x2=9, where we could take square root of each side to solve. However, if we had something like 3x=9, notice we cannot take the xth root of 9 because the index is unknown. However, we may notice that 32=9 and conclude that if 3x=32, then x=2. This is a simple example, but what if we had something a little more complex, like 10.98564x=34.9016? Then the value of x isn’t as obvious. These are the cases we address in this section and chapter.

Note

One common application of exponential functions is population growth. According to the 2009 CIA World Factbook, the country with the highest population growth rate is tied between the United Arab Emirates (north of Saudi Arabia) and Burundi (central Africa) at 3.69%. There are 32 countries with negative growth rates, the lowest being the Northern Mariana Islands (north of Australia) at 7.08%.

Definition: Exponential Function

An exponential function is a function of the form

f(x)=ax,

where f is a function of x, a>0 and a1.

Graph Exponential Functions

Let’s start to take a look at exponential functions by looking at their graphs.

Example 12.2.1

Plot f(x)=3x by plotting points. From the graph, determine the domain of the function.

Solution

Let’s pick five x-coordinates, and find corresponding y-values. Each x-value being positive or negative, and zero. This is common practice, but not required.

Table 12.2.1
x f(x)=3x (x,f(x))
2 f(2)=32=19 (2,19)
1 f(1)=31=13 (1,13)
0 f(0)=30=1 (0,1)
1 f(1)=31=3 (1,3)
2 f(2)=32=9 (2,9)

Plot the five ordered-pairs from the table. To connect the points, be sure to connect them from smallest x-value to largest x-value, i.e., left to right. Notice this graph is rising left to right, but, as the graph shoots to (to the left), it never touches the x-axis or intersects it, resulting in a horizontal asymptote at y=0. Since we see there are no restrictions to the graph, the domain is all real numbers or (,).

clipboard_e142dcfb3aff73a4d2e909cfabf3cc321.png
Figure 12.2.1
Properties of the Exponential Function

Property 1. The domain of an exponential function is all real numbers, i.e., (,).

Property 2. There are no x-intercepts; the y-intercept is at (0,1).

Property 3. If a>1, then the function is an increasing function. If 0<a<1, then the function is a decreasing function.

Property 4. There is a horizontal asymptote at y=0, unless there is a vertical shift.

Note

An exponential function never crosses the x-axis. In fact, the general exponential function isn’t defined at f(x)=0. Take a look. If f(x)=0, then f(x)=0=ax. Ask, “For which value(s) of x such that a is raised to the power of x and the result is zero?” There exists no such x. We cannot raise a positive real number to a power and the result be zero. In the event an exponential function crosses the x-axis, then that means there was a transformation to the general exponential function.

Example 12.2.2

Plot f(x)=(13)x by plotting points. From the graph, determine the domain of the function.

Solution

Let’s pick five x-coordinates, and find corresponding y-values. Each x-value being positive or negative, and zero. This is common practice, but not required.

Table 12.2.2
x f(x)=13x (x,f(x))
2 f(2)=(13)2=9 (2,9)
1 f(1)=(13)1=3 (1,3)
0 f(0)=(13)0=1 (0,1)
1 f(1)=(13)1=13 (1,13)
2 f(2)=(13)2=19 (2,19)

Plot the five ordered-pairs from the table. To connect the points, be sure to connect them from smallest x-value to largest x-value, i.e., left to right. Notice this graph is falling left to right, but, as the graph shoots to (to the right), it never touches the x-axis or intersects it. Since we see there are no restrictions to the graph, the domain is all real numbers or (,), and there is a horizontal asymptote at y=0.

clipboard_ef06c3644dd6a8e42b15d99d9927321cb.png
Figure 12.2.2

Exponential Equations with a Common Base

Since the exponential function is one-to-one, we get the following.

Solving Exponential Equations with a Common Base

To solve an exponential equation with a common base on each side of the equation, we use the fact that if

am=an, then m=n.

Example 12.2.3

Solve the equation: 52x+1=125

Solution

We use the fact above to solve the equation.

52x+1=125Rewrite 125 as 5352x+1=53Common base, equate exponents2x+1=3Solve for x2x=2Divide both sides by 2x=1Solution

We can always check the answer by verifying the solution.

52x+1?=125Plug-n-chug x=152(1)+1?=125Simplify the left side53?=125Evaluate 53125=125 True

Since we obtain a true statement by verifying the solution, then x=1 is the solution.

Example 12.2.4

Solve the equation: 83x=32

Solution

In this case, it may not seem as obvious at first, but if we rewrite each base as a common base, then we can apply the fact. Let’s rewrite each base as a common base of 2.

83x=32Rewrite 8 as 23 and 32 as 25(23)3x=25Multiply exponents 3 and 3x29x=25Common base, equate exponents9x=5Solve for xx=59Solution

We can always verify the solution, but we leave this to the student.

Example 12.2.5

Solve the equation: (19)2x=37x1

Solution

In this case, it may not seem as obvious at first, but if we rewrite each base as a common base, then we can apply the fact. Let’s rewrite each base as a common base of 3.

(19)2x=37x1Rewrite 19 as 132(132)2x=37x1Rewrite 132 as 32(32)2x=37x1Multiply exponents 2 and 2x34x=37x1Common base, equate exponents4x=7x1Solve11x=1Isolate xx=111Solution

We can always verify the solution, but we leave this to the student.

Example 12.2.6

Solve the equation: 54x52x1=53x+11

Solution

In this case, it may not seem as obvious at first, but we need to apply the product rule of exponents and obtain only one common base on each side of the equation in order to apply the fact.

54x52x1=53x+11Apply product rule of exponents on the left side54x+2x1=53x+11Simplify the exponent on the left side56x1=53x+11Common base, equate exponents6x1=3x+11Combine like terms3x=12Isolate xx=4Solution

We can always verify the solution, but we leave this to the student.

Notice, the examples only present a technique for solving exponential equations with a common base. However, not all exponential equations are written with a common base. For example, something like 2=10x cannot be written with a common base. To solve problems where we cannot rewrite the bases with a common base, we need the logarithmic function, which we will discuss in the next section.

Exponential Functions Homework

Graph each exponential function.

Exercise 12.2.1

f(x)=4x

Exercise 12.2.2

x(y)=(14)y

Exercise 12.2.3

f(x)=3x

Exercise 12.2.4

q(r)=(15)r

Exercise 12.2.5

h(n)=(12)n

Exercise 12.2.6

g(x)=2x

Exercise 12.2.7

j(x)=2x

Exercise 12.2.8

k(t)=(12)t

Solve the equation.

Exercise 12.2.9

312n=313n

Exercise 12.2.10

42a=1

Exercise 12.2.11

(125)k=1252k2

Exercise 12.2.12

62m+1=136

Exercise 12.2.13

63x=36

Exercise 12.2.14

64b=25

Exercise 12.2.15

(14)x=16

Exercise 12.2.16

43a=43

Exercise 12.2.17

363x=2162x+1

Exercise 12.2.18

92n+1=243

Exercise 12.2.19

33x2=33x+1

Exercise 12.2.20

32x=33

Exercise 12.2.21

5m+2=5m

Exercise 12.2.22

(136)b1=216

Exercise 12.2.23

622x=62

Exercise 12.2.24

423n1=14

Exercise 12.2.25

43k3422k=16k

Exercise 12.2.26

92x(1243)3x=243x

Exercise 12.2.27

64n216n+2=(14)3n1

Exercise 12.2.28

53n352n=1

Exercise 12.2.29

42x=116

Exercise 12.2.30

163p=643p

Exercise 12.2.31

625n2=1125

Exercise 12.2.32

62r3=6r3

Exercise 12.2.33

52n=5n

Exercise 12.2.34

2163v=363v

Exercise 12.2.35

272n1=9

Exercise 12.2.36

43v=64

Exercise 12.2.37

64x+2=16

Exercise 12.2.38

162k=164

Exercise 12.2.39

243p=273p

Exercise 12.2.40

42n=423n

Exercise 12.2.41

6252x=25

Exercise 12.2.42

2162n=36

Exercise 12.2.43

(14)3v2=641v

Exercise 12.2.44

21662a=63a

Exercise 12.2.45

322p28p=(12)2p

Exercise 12.2.46

32m33m=1

Exercise 12.2.47

32x33x=1

Exercise 12.2.48

43r43r=164


This page titled 12.2: Exponential functions is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Darlene Diaz (ASCCC Open Educational Resources Initiative) via source content that was edited to the style and standards of the LibreTexts platform.

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