Introductory maths analysis chapter 04 official

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 4 Chapter 4 Exponential and Logarithmic Functions Exponential and Logarithmic Functions


INTRODUCTORY MATHEMATICAL ANALYSIS

0. Review of Algebra

1. Applications and More Algebra

2. Functions and Graphs

3. Lines, Parabolas, and Systems

4. Exponential and Logarithmic Functions

5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability

10. Limits and Continuity

11. Differentiation

12. Additional Differentiation Topics

13. Curve Sketching

14. Integration

15. Methods and Applications of Integration

16. Continuous Random Variables

17. Multivariable Calculus

INTRODUCTORY MATHEMATICAL ANALYSIS


• To introduce exponential functions and their applications.

• To introduce logarithmic functions and their graphs.

• To study the basic properties of logarithmic functions.

• To develop techniques for solving logarithmic and exponential equations.

Chapter 4: Exponential and Logarithmic Functions

Chapter ObjectivesChapter Objectives


Exponential Functions

Logarithmic Functions

Properties of Logarithms

Logarithmic and Exponential Equations

4.1)

4.2)

4.3)

4.4)


Chapter OutlineChapter Outline


• The function f defined by

where b > 0, b 1, and the exponent x is any real number, is called an exponential function with base b1.


4.1 Exponential Functions4.1 Exponential Functions xbxf


The number of bacteria present in a culture after t minutes is given by .

a. How many bacteria are present initially?

b. Approximately how many bacteria are present after 3 minutes?

Solution:

a. When t = 0,

b. When t = 3,


4.1 Exponential Functions

Example 1 – Bacteria Growth

t

tN

3

4200

04

(0) 300 300(1) 3003

N

34 64 6400

(3) 300 300 7113 27 9

N


Graph the exponential function f(x) = (1/2)x.

Solution:



Example 3 – Graphing Exponential Functions with 0 < b < 1


Properties of Exponential Functions




Solution:

Compound Interest

• The compound amount S of the principal P at the end of n years at the rate of r compounded annually is given by

.



Example 5 – Graph of a Function with a Constant Base2

Graph 3 .xy

(1 )nS P r




Example 7 – Population Growth

The population of a town of 10,000 grows at the rate of 2% per year. Find the population three years from now.

Solution:

For t = 3, we have .3(3) 10,000(1.02) 10,612P




Example 9 – Population Growth

The projected population P of a city is given by where t is the number of years after

1990. Predict the population for the year 2010.

Solution:

For t = 20,0.05(20) 1100,000 100,000 100,000 271,828P e e e

0.05100,000 tP e




Example 11 – Radioactive Decay

A radioactive element decays such that after t days the number of milligrams present is given by

.

a. How many milligrams are initially present?

Solution: For t = 0, .

b. How many milligrams are present after 10 days?

Solution: For t = 10, .

0.062100 tN e

mg 100100 0062.0 eN

mg 8.53100 10062.0 eN



4.2 Logarithmic Functions4.2 Logarithmic Functions

Example 1 – Converting from Exponential to Logarithmic Form

• y = logbx if and only if by=x.

• Fundamental equations are andlogb xb xlog x

b b x

25

4

a. Since 5 25 it follows that log 25 2b. Since 3 81 it follo

Exponential Form Logarithmic Form 3

010

ws that log 81 4c. Since 10 1 it follows that log 1 0



4.2 Logarithmic Functions

Example 3 – Graph of a Logarithmic Function with b > 1

Sketch the graph of y = log2x.

Solution:




Example 5 – Finding Logarithms

a. Find log 100.

b. Find ln 1.

c. Find log 0.1.

d. Find ln e-1.

d. Find log366.

210log100log 2

01ln

110log1.0log 1

1ln1ln 1 ee

2

1

6log2

6log6log36




Example 7 – Finding Half-Life

• If a radioactive element has decay constant λ, the half-life of the element is given by

A 10-milligram sample of radioactive polonium 210 (which is denoted 210Po) decays according to the equation. Determine the half-life of 210Po.

Solution:

2ln

T

daysλ

T 4.13800501.0

2ln2ln



4.3 Properties of Logarithms4.3 Properties of Logarithms

Example 1 – Finding Logarithms

• Properties of logarithms are:

nmmn bbb loglog)(log .1

nmn

mbb logloglog .2 b

mrm br

b loglog 3.

a.

b.

c.

d.

7482.18451.09031.07log8log)78log(56log

6532.03010.09542.02log9log2

9log

8062.1)9031.0(28log28log64log 2 3495.0)6990.0(

2

15log

2

15log5log 2/1

b

mm

b

mm

a

ab

b

b

bb

log

loglog .7

1log .6

01log .5

log1

log 4.



4.3 Properties of Logarithms

Example 3 – Writing Logarithms in Terms of Simpler Logarithms

a.

b.

wzx

wzx

zwxzw

x

lnlnln

)ln(lnln

)ln(lnln

)]3ln()2ln(8ln5[3

1

)]3ln()2ln([ln3

1

)}3ln(])2({ln[3

1

3

)2(ln

3

1

3

)2(ln

3

)2(ln

85

85

853/185

3

85

xxx

xxx

xxx

x

xx

x

xx

x

xx




Example 5 – Simplifying Logarithmic Expressions

a.

b.

c.

d.

e.

.3ln 3 xe x

3

30

10log01000log1log 3

989/8

79 8

7 7log7log

1)3(log3

3log

81

27log 1

34

3

33

0)1(1

10logln10

1logln 1

ee




Example 7 – Evaluating a Logarithm Base 5

Find log52.

Solution:

4307.05log

2log

2log5log

2log5log

25

x

x

x

x




4.4 Logarithmic and Exponential Equations4.4 Logarithmic and Exponential Equations

• A logarithmic equation involves the logarithm of an expression containing an unknown.

• An exponential equation has the unknown appearing in an exponent.


An experiment was conducted with a particular type of small animal. The logarithm of the amount of oxygen consumed per hour was determined for a number of the animals and was plotted against the logarithms of the weights of the animals. It was found that

where y is the number of microliters of oxygen consumed per hour and x is the weight of the animal (in grams). Solve for y.


4.4 Logarithmic and Exponential Equations

Example 1 – Oxygen Composition

xy log885.0934.5loglog


Solution:



Example 1 – Oxygen Composition

)934.5log(log

log934.5log

log885.0934.5loglog

885.0

885.0

xy

x

xy

885.0934.5 xy




Example 3 – Using Logarithms to Solve an Exponential Equation

Solution:

.124)3(5 1 xSolve

61120.1

ln4ln

4

124)3(5

371

371

1

x

x

x

x


In an article concerning predators and prey, Holling refers to an equation of the form where x is the prey density, y is the number of prey attacked, and K and a are constants. Verify his claim that

Solution:Find ax first, and thus



Example 5 – Predator-Prey Relation

axyK

K

ln

)1( axeKy

K

yKe

eK

y

eKy

ax

ax

ax

1

)1(

axyK

K

axK

yK

axK

yK

ln

ln

ln

(Proved!)

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