chapter 4 - exponential and logarithmic functions

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 Functions5. Mathematics of Finance

6. Matrix Algebra

7. Linear Programming

8. Introduction to Probability and Statistics


9. Additional Topics in Probability10. Limits and Continuity11. Differentiation12. Additional Differentiation Topics13. Curve Sketching14. Integration15. Methods and Applications of Integration16. Continuous Random Variables17. 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,

Chapter 4: Exponential and Logarithmic Functions4.1 Exponential Functions

Example 1 – Bacteria Growth

t

tN

34200

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 GrowthThe 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 DecayA 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 and

logb xb xlog xb 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


Chapter 4: Exponential and Logarithmic Functions4.2 Logarithmic Functions

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

Sketch the graph of y = log2x. Solution:



Example 5 – Finding Logarithmsa. 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

21

6log26log6log36



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

nmnm

bb logloglog .2 b

mrm br

b loglog 3.

a.b.c.d.

7482.18451.09031.07log8log)78log(56log

6532.03010.09542.02log9log29log

8062.1)9031.0(28log28log64log 2 3495.0)6990.0(

215log

215log5log 2/1

bmm

b

mm

a

ab

b

b

bb

logloglog .7

1log .601log .5

log1log 4.


Chapter 4: Exponential and Logarithmic Functions4.3 Properties of Logarithms

Example 3 – Writing Logarithms in Terms of Simpler Logarithmsa.

b.

wzxwzx

zwxzwx

lnlnln)ln(lnln

)ln(lnln

)]3ln()2ln(8ln5[31

)]3ln()2ln([ln31

)}3ln(])2({ln[31

3)2(ln

31

3)2(ln

3)2(ln

85

85

853/1853

85

xxx

xxx

xxx

xxx

xxx

xxx



Example 5 – Simplifying Logarithmic Expressionsa. b.

c.

d.

e.

.3ln 3 xe x

330

10log01000log1log 3

989/8

79 8

7 7log7log

1)3(log33log

8127log 1

34

3

33

0)1(1

10logln101logln 1

ee



Example 7 – Evaluating a Logarithm Base 5Find log52.

Solution:

4307.05log2log2log5log2log5log

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.

Chapter 4: Exponential and Logarithmic Functions4.4 Logarithmic and Exponential Equations

Example 1 – Oxygen Composition

xy log885.0934.5loglog


Solution:

Chapter 4: Exponential and Logarithmic Functions4.4 Logarithmic and Exponential EquationsExample 1 – Oxygen Composition

)934.5log(log

log934.5log

log885.0934.5loglog

885.0

885.0

xyx

xy

885.0934.5 xy



Example 3 – Using Logarithms to Solve an Exponential Equation

Solution:

.124)3(5 1 xSolve

61120.1ln4ln

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

KyKe

eKy

eKy

ax

ax

ax

1

)1(

axyK

K

axK

yK

axK

yK

ln

ln

ln

(Proved!)