© 2008 Pearson Addison-Wesley. All rights reserved 8-6-1 Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models

© 2008 Pearson Addison-Wesley. All rights reserved

8-6-1

Chapter 1

Section 8-6Exponential and Logarithmic

Functions, Applications, and Models


8-6-2

Exponential and Logarithmic Functions, Applications, and Models

• Exponential Functions and Applications

• Logarithmic Functions and Applications

• Exponential Models in Nature


8-6-3

Exponential Function

( ) ,xf x b

An exponential function with base b, where b > 0 and is a function of the form

where x is any real number.

1,b


8-6-4

Example: Exponential Function (b > 1)

y

x

( ) 2xf x

The x-axis is the horizontal asymptote of each graph.

x 2 x

–1 1/2

0 1

1 2

2 4


8-6-5

Example: Exponential Function (0 < b < 1)

y

x

1( )

2

x

f x

The x-axis is the horizontal asymptote of each graph.

x (1/2) x

–2 4

–1 2

0 1

1 1/2


8-6-6

Graph of ( ) xf x b

1. The graph always will contain the point (0, 1).2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right.3. The x-axis is the horizontal asymptote.4. The domain is and the range is( , ) (0, ).


8-6-7

Exponential with Base e

2.718281828.e

y

x

( ) xf x e


8-6-8

Compound Interest Formula

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

1 .nt

rA P

n


8-6-9

Example: Compound Interest Formula

Suppose that $2000 dollars is invested at an annual rate of 8%, compounded quarterly. Find the total amount in the account after 6 years if no withdrawals are made.

Solution1

ntr

A Pn

4(6).08

2000 14

A


8-6-10


Solution (continued)

242000 1.02A

2000 1.60844 3216.88A

There would be $3216.88 in the account at the end of six years.


8-6-11

Continuous Compound Interest Formula

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

.rtA Pe


8-6-12

Example: Continuous Compound Interest Formula

Suppose that $2000 dollars is invested at an annual rate of 8%, compounded continuously. Find the total amount in the account after 6 years if no withdrawals are made.

Solution

.08(6)2000A e

rtA Pe


8-6-13


Solution (continued)

There would be $3232.14 in the account at the end of six years.

.482000A e2000(1.61607) 3232.14A


8-6-14

Exponential Growth Formula

The continuous compound interest formula is an example of an exponential growth function. In situations involving growth or decay of a quantity, the amount present at time t can often be approximated by a function of the form

0( ) ,ktA t A ewhere A0 represents the amount present at time t = 0, and k is a constant. If k > 0, there is exponential growth; if k < 0, there is exponential decay.


8-6-15

Definition of

then log .ybb x x

logb x

For b > 0, 1, if b


8-6-16

Exponential and Logarithmic Equations

Exponential Equation

Logarithmic Equation

43 81 34 log 81410 10,00034 1/ 64 03 1

104 log 10,000

43 log (1/ 64)

30 log 1


8-6-17

Logarithmic Function

A logarithmic function with base b, where b > 0 and is a function of the form

1,b

( ) log , where 0.bg x x x


8-6-18

Graph of ( ) logbg x x

The graph of y = log b x can be found by interchanging the roles of x and y in the function f (x) = bx. Geometrically, this is accomplished by reflecting the graph of f (x) = bx about the line y = x.

The y-axis is called the vertical asymptote of the graph.


8-6-19

Example: Logarithmic Functions

y

x

y

x

2( ) logg x x 1/ 2( ) logg x x


8-6-20

Graph of ( ) logbg x x

1. The graph always will contain the point (1, 0).2. When b > 1 the graph will rise from left to right. When 0 < b < 1, the graph will fall from left to right.3. The y-axis is the vertical asymptote.4. The domain is and the range is ( , ). (0, )


8-6-21

Natural Logarithmic Function

ln logex xg(x) = ln x, called the natural logarithmic function, is graphed below.

y

x

( ) lng x x


8-6-22

Natural Logarithmic Function

ln .ke k

The expression ln ek is the exponent to which the base e must be raised in order to obtain ek. There is only one such number that will do this, and it is k. Thus for all real numbers k,


8-6-23

Example: Doubling Time

Suppose that a certain amount P is invested at an annual rate of 5% compounded continuously. How long will it take for the amount to double (doubling time)?

SolutionrtA Pe.052 tP Pe

.052 te

Sub in 2P for A (double).

Divide by P.


8-6-24

Example: Doubling Time

Solution (continued).05ln 2 ln te

ln 2 .05tln 2

13.9.05

t

Therefore, it would take about 13.9 years for the initial investment P to double.

Divide by .05

Take ln of both sides.

Simplify.


8-6-25

Exponential Models in Nature

Radioactive materials disintegrate according to exponential decay functions. The half-life of a quantity that decays exponentially is the amount of time it takes for any initial amount to decay to half its initial value.


8-6-26

Example: Half-Life

Carbon 14 is a radioactive form of carbon that is found in all living plants and animals. After a plant or animal dies, the radiocarbon disintegrates. The amount of carbon 14 present after t years is modeled by the exponential equation

.00012160

ty y ea) What is the half-life of carbon 14?b) If an initial sample contains 1 gram of carbon 14, how much will be left in 10,000 years?


8-6-27

Example: Half-Life

Solution.0001216

0ty y ea)

.00012160 0

1

2ty y e

1ln .0001216

2t

5700t The half-life of carbon 14 is about 5700 years.


8-6-28

Example: Half-Life

Solution.0001216(10000)1y eb)

.30y

There will be about .30 grams remaining.

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© 2008 Pearson Addison-Wesley. All rights reserved 8-6-1 Chapter 1 Section 8-6 Exponential and Logarithmic Functions, Applications, and Models