Exponential Functions. Exponential Functions and Their Graphs. Example 2. Example 1. Example 3. Graphs of Exponential Functions 9/17/2014

Exponential Functions and Their

Graphs

Precalculus 3.1

Exponential Functions

Example 1

Use a calculator to evaluate each function at the indicated value of x. a) x= b) x=1/2 c) x=-2.5 x x f( )8 x x f( ) 8 x x f( )0.8 

Example 2

In the same coordinate place, sketch the graph of each function. a) x b) x f( )3 x x f( )5

Example 3

In the same coordinate place, sketch the graph of each function.

a) x b)

x

f( ) 3 f(x) 5x

Graphs of Exponential Functions

• The basic characteristics of exponential functions y = ax

and y = a–x_{are summarized in Figures 3.3 and 3.4.}

• Graph of y = ax_{, a > 1}

• Domain: ( , ) • Range: (0, ) • y-intercept: (0, 1) • Increasing

• x-axis is a horizontal asymptote (ax_{→ 0, as x→ ).} Figure 3.3

Graphs of Exponential Functions

• Graph of y = a–x_{, a > 1} • Domain: ( , ) • Range: (0, ) • y-intercept: (0, 1) • Decreasing

• x-axis is a horizontal asymptote (a–x_{→ 0, as x→ ).}

• Continuous

• From Figures 3.3 and 3.4, you can see that the graph of an exponential function is always increasing or always decreasing.

Figure 3.4

Graphs of Exponential Functions

• As a result, the graphs pass the Horizontal Line

Test, and therefore the functions are one-to-one functions.

• You can use the following One-to-One Property to solve simple exponential equations.

• For a > 0 and a ≠ 1, ax_{= a}y_{if and only if x = y.} One-to-One Property

Example 4

Solve. a) 2 b) 2 16 x

 

3 81 1 x_

Example 5

Describe the graph as a transformation of the graph of a) b) c) x x f( )4 2 4 ) (   x x f

 

x x f( ) 2 4 1   3 4 ) (x   x f

The Natural Base e

• In many applications, the most convenient choice for a base is the irrational number

• e  2.718281828 . . . .

• This number is called the natural base. • The function given by f(x) = ex_is • called the natural exponential

Example 6

Use a calculator to evaluate the function given by at each value of x to three decimal places. a) x = 6.2 b) x = -0.4 c) x = -7.1 d) x= 0.72 x e x f( )

Example 7

Sketch the graph of the function s( t) 5e0.17t

Applications

Example 8

On the day of a child’s birth, a deposit of $25,000 is made in a trust fund that pays 8.25% interest. Determine the balance in this account on the child’s 26th_{birthday if the}

interest is compounded

a)quarterly b) monthly c) continuously

Example 9

Let Q represent the mass of radium

whose half-life is 1620 years. The quantity of radium present after t years is given by a) Sketch the graph of the function over the

interval from t=0 to t=5000.

b) Determine the initial quantity (when t=0) c) Determine the quantity present after 1000

years.

 

/1620 2 1 16 t Q  Ra 226

Logarithmic Functions and Their

Graphs

Precalculus 3.2

Logarithmic Functions

• Every function of the form f (x) = ax _{passes the}

Horizontal Line Test and therefore must have an inverse function. This inverse function is called the logarithmic function with base a.

Example 1

Use the definition of logarithmic function to evaluate each logarithm at the indicated value of x. a) b) c) d) 16 , log ) (x  ₄x x f 64 , log ) (x  ₂x x f 1 , log ) (x  ₅x x f 81 1 3 , log ) (x  x x f

Logarithmic Functions

• The logarithmic function with base 10 is called the common logarithmic function. It is denoted by log₁₀or simply by log. On most calculators, this function is denoted by .

Example 2

Use a calculator to evaluate the function at each value of x to three decimal places.

a)x = 100 b) x = 1/5 c) x = 3.25 d) -4 x x f( )log

Logarithmic Functions

Example 3

Simplify. a) b) c) 1 log₅ log 11 11 30 log8 8

Example 4

Solve. a) b) c) 16 log

log₅y ₅ log(43x)log(x2)

29 log ) 4 ( log₃ x2  ₃

Example 5

In the same coordinate plane, sketch the graph of each function. a) b) x x f( )4 x x f( )log₄

Graphs of Logarithmic Functions

• The basic characteristics of logarithmic graphs are summarized in Figure 3.16. Graph of y = logax, a  1 • Domain: (0, ) • Range: ( , ) • x-intercept: (1, 0) • Increasing

• One-to-one, therefore has an inverse function

Figure 3.16

Example 6

Sketch the graph of

Identify the vertical asymptote.

 

4 log ) (_x x f 

Example 7

Describe the graph as a transformation of the graph of a) b) x x f( )log3 1 log ) (x  3x f f(x)log3(x2)

The Natural Logarithmic Function

Example 8

Use a calculator to evaluate the function to three decimal places.

a) x = 73.25 b) x = 0.4 c) x = -2 d) x = 1 ln ) (x  x f 3 2 

The Natural Logarithmic Function

Example 9

Use the properties of natural logarithms to simplify. a) b) c) d) 3 / 1 ln e _eln8 1 ln 15 6 ln e

Example 10

Find the domain of each function.

a) b) c) ) 3 ln( ) (x  x f f(x)ln(3x) 3 ln ) (x x f 

Example 11

Students in a mathematics class were given an exam and then retested monthly with an equivalent exam. The average scores for the class are given by the human memory model where t is time in months.

a) What was the average score on the original exam? (t=0) 12 0 ), 1 log( 17 78 ) (t   t t f

Example 11

b) What was the average score after 3 months? c) What was the average score after 11 months?

12 0 ), 1 log( 17 78 ) (t   t t f

Properties of Logarithms

Precalculus 3.3

Change of Base

• Most calculators have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e).

• Although common logarithms and natural logarithms are the most frequently used, you may occasionally need to evaluate logarithms with other bases. To do this, you can use the following change-of-base formula.

Change of Base

Example 1

Evaluate each using the change-of-base formula with common logs. Approximate to three decimal places.

a)log₃16 b)log₅22

Example 2

Evaluate each using the change-of-base formula with natural logs. Approximate to three decimal places.

a)log₃16 b)log₅22

Properties of Logarithms

Example 3

Write each logarithm in terms of ln2 and ln5. a) b) 10 ln 2 5 ln

Example 4

Find the exact value of each expression without using a calculator.

a) 5 b)

7 7

log lne 12 lne5

Example 5

Expand each logarithmic expression.

a)_log3x2y b) 8 1 4 ln x

Example 6

Condense each logarithmic expression. a) b) c) ) 3 log( 5 log 3 1 _{x x} x x 4) 2ln ln( 4  



log₃ log₃( 2)



5 1 _{x x}

Exponential & Logarithmic

Equations

Precalculus 3.4

Example 1

Solve. a) b) c) d) e) f) 512 2 x _{ln5lnx0}

 

₅1 x_125 _ex_13 8 lnx logx2

Example 2

Solve each equation and approximate the result to three decimal places if necessary.

a)_ex2 _e5x6 b)₄

 

₃x _64

Example 3

Solve. 53ex2

Example 4

Solve the equation and approximate the result to three decimal places.



2



4 11 6 t5 _{ }

Example 5

Solve. _e2x_7_ex_12_0

Example 6

Solve. a) b) c) 3 2 ln x log₄(3x2)log₄(6x) x x 13) log 6 log 3 5 ( log₃   ₃  ₃

Example 7

Solve the equation and approximate the result to three decimal places.

4 ln 3

Example 8

Solve.3log₄6x9

Example 9

Solve. log₁₀xlog₁₀(x9)1

Example 10

You have deposited $1000 in an account that pays 6.25% compounded continuously. How long will it take your money to double?

Example 11

The number y of endangered animal species on a protected wildlife preserve from 1990 to 2004 can be modeled by

where t represents the year, with t=10 corresponding to 1990. During which year did the number of endangered animal species reach 342?

t y117 159ln ,

24 10 t

Exponential & Logarithmic

Models

Precalculus 3.5

Introduction

• The five most common types of mathematical models involving exponential functions and logarithmic functions are as follows. • 1. Exponential growth model: y = aebx_{, b  0} • 2. Exponential decay model: y = ae–bx_{, b  0}

Introduction

• The basic shapes of the graphs of these functions are shown in Figure 3.33.

Exponential growth model Exponential decay model Gaussian model

Figure 3.33

Introduction

Logistic growth model Natural logarithmic model Common logarithmic model

Figure 3.33

Example 1

The population P of a city is given by

where t=0 represents 2001. According to this model, when did the population reach 150,000?

t

e P_95,300 0.055

Example 2

In a research experiment, a population of fruit flies is increasing according to the law of exponential growth. After 2 days there are 125 fruit flies, and after 4 days there are 350 flies. How many flies will there be after 6 days?

Example 3

Estimate the age of a newly discovered fossil in which the ratio of carbon 14 to carbon 12 is

14 10 1  R

Gaussian Models

• The Gaussian models are of the form • y = ae–(x – b)2_/c

.

• This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve.

• The average value of a population can be found from the bell-shaped curve by observing where the maximum

y–value of the function occurs.

• The x–value corresponding to the maximum y– value of the

function represents the average value of the independent

variable—in this case, x.

Example 4

Last year, the math scores for students in a particular math class roughly followed the normal distribution given by

where x is the math score. Sketch the graph of this function, and use it to estimate the average math score.

110 30 , 0399 . 0 114 2 ) 74 (    e  x y x

Logistic Growth Models

• Some populations initially have rapid growth, followed by a

declining rate of growth, as indicated by the graph in Figure 3.40.

• One model for describing this type of growth pattern is the logistic curve given by the function

• where y is the population size and

x is the time.

Figure 3.40

Logistic Growth Models

• An example is a bacteria culture that is initially

allowed to

grow under ideal conditions, and then under less favorable

conditions that inhibit growth. • A logistic growth curve is also called a

sigmoidal curve.

Example 5

On a college campus of 7500 students, one student returns from vacation with a contagious and long-lasting virus. The spread of the virus is modeled by

where y is the total number of students affected 0 , 7499 1 7500 9 . 0    _ t e y _t

Example 5

a) How many students will be infected after 4 days?

b) After how many days will the college cancel classes? 0 , 7499 1 7500 9 . 0    _ t e y _t

Example 6

On the Richter scale, the magnitude R of an earthquake of intensity is given by

where is the minimum intensity used for comparison. Find the magnitude R of an earthquake of intensity a) b) 0 log I I R  I 1 0 I . I 000 , 400 , 68  I I42,275,000