Copyright © 2007 Pearson Education, Inc. Slide 5-1.

Copyright © 2007 Pearson Education, Inc. Slide 5-1


Chapter 5: Inverse, Exponential and Logarithmic Functions

5.1 Inverse Functions

5.2 Exponential Functions

5.3 Logarithms and Their Properties

5.4 Logarithmic Functions

5.5 Exponential and Logarithmic Equations and Inequalities

5.6 Further Applications and Modeling with Exponential and Logarithmic Functions


5.2 Exponential Functions

• Additional Properties of Exponents– ax is a unique real number.

– ab = ac if and only if b = c.

– If a > 1 and m < n, then am < an.

– If 0 < a < 1 and m < n, then am > an.

If a > 0, a 1, then

f (x) = ax

is the exponential function with base a.

32 44 e.g. 3

212

21 e.g.


5.2 Graphs of Exponential Functions

Example Graph Determine the domain and range of f.

Solution

There is no x-intercept. Any number to the zero power is 1, so the y-intercept is (0,1). The domain is (– ,), and the range is (0,).

.2)( xxf


5.2 Graph of f (x) = ax, a > 1


5.2 Graph of f (x) = ax, 0 < a < 1


5.2 Comparing Graphs

Example Explain why the graph ofis a reflection across the y-axis of the graph of

Analytic SolutionShow that g(x) = f (–x).

xxg 21)(

.2)( xxf

)(2

2

21

)(

1

xf

xg

x

x

x


5.2 Comparing Graphs

Graphical Solution

The graph below indicates that g(x) is a reflection

across the y-axis of f (x).


5.2 Using Translations to Graph an Exponential Function

Example Explain how the graph of is obtained from the graph of

Solution

2 3 xy

.2xy

units 3 up d translateis 2 32

axis- theacross 2reflect 2xx

xx

y

xy


5.2 Example using Graphs to Evaluate Exponential Expressions

Example Use a graph to evaluate

Solution With we find

that y 2.6651441 from the graph of y = .5x.

.5. 2

,414214.12 x


5.2 Exponential Equations (Type I)

ExampleSolve

Solution

.12525 x

12525 x

Write with the same base. 32 55 x

mnnm aa 32 55 x

Set exponents equal and solve.

2332

x

x

. isset solution The.12525 :Verify 232

3


5.2 Using a Graph to Solve Exponential Inequalities

Example Solve the inequality

Solution Using the graph below, the graph lies above the x-axis for values of x less than .5.

.05.1 8271 xx

The solution set for y > 0 is (–,.5).


5.2 The Natural Number e

• Named after Swiss mathematician Leonhard Euler

• e involves the expression

• e is an irrational number• Since e is an important

base, calculators are programmed to find powers of e.

:1 1 xx

718281828.2 1, as 1 ex xx


5.2 Compound Interest

• Recall simple earned interest where– P is the principal (or initial investment), – r is the annual interest rate, and – t is the number of years.

• If A is the final balance at the end of each year, then

,tPrI

nth rPAn

rPrrPrrPrPArPPrPA

)1(:year

)1()1)(1()1()1(:year 2)1(:year 1

2nd

st


5.2 Compound Interest Formula

Example Suppose that $1000 is invested at an annual rate of 8%, compounded quarterly. Find the total amount in the account after 10 years if no withdrawals are made.

Solution

The final balance is $2208.04.

Suppose that a principal of P dollars is invested at an annual interest rate r (in percent), compounded n times per year. Then, the amount A accumulated after t years is given by the formula

.1nt

nr

PA

039664.2208408.

11000104

A


5.2 Continuous Compounding Formula

Example Suppose $5000 is deposited in an account paying 8% compounded continuously for 5 years. Find the total amount on deposit at the end of 5 years.

Solution

The final balance is $7459.12.

If P dollars is deposited at a rate of interest r compounded continuously for t years, the final amount A in dollars on deposit is

.rtPeA

12.745950005000 4.)5(08. eeA


5.2 Modeling the Risk of Alzheimer’s Disease

Example The chances of dying of influenza or pneumonia increase exponentially after age 55 according to the function defined by

where r is the risk (in decimal form) at age 55 and x is the number of years greater than 55. What is the risk at age 75?

Solution x = 75 – 55 = 20, soThus, the risk is almost fives times as great at age 75

as at age 55.

( ) (1.082) , xf x r

20( ) (1.082) 4.84 f x r r

Copyright © 2007 Pearson Education, Inc. Slide 5-1.

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