BUSMATH(1)

BUSMATH (Business Math)


Kristine Joy E. Carpio

Department of MathematicsDe La Salle University – Manila

Term 2 2011-2012


Outline

Exponential and Logarithmic FunctionsExponential Functions and Their ApplicationsLogarithmic Functions and Their ApplicationsProperties of LogarithmsExponential and Logarithmic Equations

Geometric Sequence and SeriesGeometric SequenceGeometric Series

References


Course Description

This course covers the theory of investment mathematics andtheir application to commerce and economics. This course alsointroduces students to the rudiments of the mathematics offinance. Special topics in College Algebra that are requisiteknowledge in solving investment problems will also be reviewedsuch as exponential and logarithmic functions and geometricsequence.


Exponential and Logarithmic Functions

Exponential Functions and Their Applications


DefinitionAn exponential function is a function of the form

f (x) = bx,

where b > 0 and b 6= 1.

For many applications the base 10 or e is used. The number e isan irrational number that is approximately 2.718. Base 10 iscalled the common base and e is called the the natural base.




Characteristics of the exponential function:

1. The y-intercept of the curve is (0, 1).

2. The domain is (−∞, +∞) and the range is (0, +∞).

3. The curve approaches the negative x-axis when b > 1;when 0 < b < 1 the curve approaches the positive x-axis.

4. The curve is increasing from left to right when b > 1; when0 < b < 1 the curve is decreasing from left to right.




Exercises

Sketch the graph of each function.

1. y = 10x+2

2. y = −2x

3. y = 2−x

4. y = 2(x2)

5. y = (0.1)x




DefinitionIf a function is such that no two ordered pairs have differentx-coordinates and the same y-coordinate, then the function iscalled one-to-one function.

Functions that are one-to-one are invertible functions.

DefinitionThe inverse of a one-to-one function f is the function f −1, whichis obtained from f by interchanging the coordinates in eachordered pair of f .

One-to-One Property of Exponential Functions:For b > 0 and b 6= 1,

if bm = bn, then m = n

.




Exercises

Solve each equation.

1. 102x = 0.1

2. 3x =1

9

3.

(

1

4

)3x

= 16

4. 10−x = 0.01

5. −32−x = −81




Applications

Definition (Compound Interest Formula)

If P represents the principal, i (in decimal form) the annualinterest rate, m the number times the interest is compounded ina year, t the number of years, and A the amount at the end of t

years, then

A = P

(

1 +i

m

)mt

.

If the length of the time period is shortened then the number ofperiods n increases while the interest rate for the perioddecreases. As n increases, the amount A also increases but willnot exceed a certain amount. That certain amount is theamount obtained from continuous compounding of the interest.




Definition (Continuous-Compounding Formula)

If P is the principal or beginning balance, i (in decimal form) isthe annual percentage rate compounded continuously, t is thetime in years, and A is the amount of or ending balance, then

A = Peit




Exercises

Solve each problem.

1. If $6000 is deposited in an account paying 5% compoundedquarterly, then what amount will be in the account after 10years?

2. The value of a certain textbook seems to decreaseaccording to the formula V = 45 · 2−0.9t , where V is thevalue in dollars and t is the age of the book in years. Whatis the book worth when the it is new? What is it worthwhen it is 2 years old?

3. How much interest will be earned the first year on $80000on deposit in an account paying 7.5% compundedcontinuously?



Logarithmic Functions and Their Applications


Definition (logb(x))

For any b > 0 and b 6= 1,

y = logb(x) if and only if by = x.

The inverse of the base-b exponential function f (x) = bx is thebase-b logarithmic function f −1(x).

There are two bases for logarithms that are used morefrequently than others: They are 10 and e. The base-10logarithm is called the common logarithm and is usually writtenas log(x). The base-e algorithm is called the natural algorithmand is usually written as ln(x).




Exercises

Evaluate each logarithm.

1. log4(64)

2. log(1)

3. log1/3(27)

4. ln(

1e

)

5. log25(5)




The graphs of a function and its inverse function are symmetricabout the line y = x. Because the logarithmic functions areinverses of the exponential functions, their graphs are alsosymmetric about y = x. Characteristics of the logarithmicfunction:

1. The x-intercept of the curve is (1, 0).

2. The domain is (0, +∞) and the range is (−∞, +∞).

3. The curve approaches the negative y-axis when b > 1;when 0 < b < 1 the curve approaches the positive y-axis.

4. The curve is increasing from left to right when b > 1; when0 < b < 1 the curve is decreasing from left to right.




Exercises

Sketch the graph of each function.

1. y = log3(x)

2. y = log4(x)

3. y = log1/4(x)

4. y = log1/5(x)

5. y = log10(x)




Exercises

One-to-One Property of Logarithmic Functions:For b > 0 and b 6= 1,

if logb(m) = logb(n), then m = n.

Solve each equation.

1. log(x) = −3

2. logx(36) = 2

3. log(x2) = log(9)

4. ln(2x − 3) = ln(x + 1)

5. x =

(

1

2

)−2




ExercisesSolve each problem.

1. How long does it take $5000 to grow to $10000 at 12% compoundedcontinuously?

2. An investment of $10000 in Baytex Energy in 1997 was worth $19,568

in 2002.

a) Assuming that the investment grew continuously, what wasthe annual rate?

b) If Baytex Energy continued to grow continuously at the ratefrom part a), then what would be the investment be worthin 2012?

3. The level of sound in decibels (dB) is given by the formula

L = 10 · log(I × 1012),

where I is the intensity of the sound in watts per square meter. If theintensity of the sound of a rock concert is 0.001 watt per square meterat a distance of 75 meters from the stage,then what is the level of thesound at this point from the audience?



Properties of Logarithms

Change-of-Base Formula

Let a, b and x be positive real numbers such that a 6= 1 andb 6= 1. Then loga x can be converted to a different base asfollows.

Base b Base 10 Base e

loga x =logb x

logb aloga x =

log x

log aloga x =

ln x

ln a




Exercises

Rewrite the algorithm as a ratio of (a) common logarithm and(b) natural logarithm

1. log3 x

2. log1/3 x

3. logx34

4. log7.1 x




Properties of LogarithmsIf M , N , and b are positive numbers, b 6= 1, then

1. logb(b) = 1

2. logb(1) = 0

Theorem (Product Rule for Logarithms)

If M, N , and b are positive numbers, b 6= 1, then

logb(MN ) = logb(M ) + logb(N )

Theorem (Quotient Rule for Logarithms)


logb

(

M

N

)

= logb(M ) − logb(N )




Theorem (Power Rule for Logarithms)


logb(M N ) = N · logb(M )

Theorem (Inverse Properties)


1. logb(bM ) = M

2. blogb(M) = M




ExercisesUse the properties of logarithms to rewrite and simplify thelogarithmic expression.

1. log2(42 · 34)

2. log 9300

3. ln 6e2

Find the exact value of the logarithmic expression withoutusing a calculator.

1. log51

125

2. log63√

6

3. log3 81−0.2

4. log2(−16)

5. 3 ln e4

6. ln4√

e3

7. 2 ln e6 − ln e5

8. log4 2 + log4 32




ExercisesUse the properties of logarithms to expand the expression as asum, difference, and/or constant multiple of logarithms.(Assume all varaibles are positive.)

1. log3 10z

2. log61z3

3. log 4x2y

4. ln 6√x2+1

5. log2

√xy4

z4

Condense the expression to the logarithm of a single quantity

1. ln5 8 − log5 t

2. −4 log6 2x

3. 3 log3 x + 4 log3 y − 4 log3 z

4. 2 [3 ln x − ln(x + 1) − ln(x − 1)]



Exponential and Logarithmic Equations

Exponential and Logarithmic EquationsStrategies for Solving Exponential and Logarithmic Equations

1. Rewrite the original equation in a form that allows the useof the One-to-One Properties of exponential andlogarithmic functions.

2. Rewrite an exponential equation in logarithmic form andapply the Inverse Property of logarithmic functions.

3. Rewrite an logarithmic equation in exponential form andapply the Inverse Property of exponential functions.

One-to-One Properties

ax = ay if and only if x = y

loga x = loga y if and only if x = y

Inverse Properties

aloga x = x

loga ax = x




Exercises

Solve the exponential equation algebraically.

1. e−x2= ex2−2x

2. 4ex = 91

3. 65x = 3000

4. 2x−3 = 32

5. 5(10x−6) = 7

6. e2x = 50

7. −14 + 3ex = 11

8. e2x − 5ex + 6 = 0

9. 4001+e−x = 350

10.(

4 − 2.47140

)9t= 21




Exercises

Solve the logarithmic equation algebraically.

1. ln x = 2

2. log 3z = 2

3. ln√

x − 8 = 5

4. 5 log10(x − 2) = 11

5. ln x + ln(x + 3) = 1

6. log(x − 6) = log(2x + 1)

7. log3 x + log3(x − 8) = 2


Geometric Sequence and Series

Geometric Sequence

Geometric Sequence

DefinitionA series is geometric if the ratios of consecutive terms are thesame. So, the sequence a1, a2, a3, a4, . . . , an , . . . is geometric ifthere is a number r such that

a2

a1= r ,

a3

a2= r ,

a4

a3= r , r 6= 0

and so the number r is the commone ratio of the sequence.



Geometric Sequence

Exercises

Determine whether the sequence is geometric. If so, find thecommon ratio.

1. 3, 12, 48, 192, . . .

2. 36, 27, 18, 9, . . .

3. 5, 1, 0.2, 0.04, . . .

4. 9, −6, 4, −83 , . . .

5. 15 ,

27 ,

39 ,

49 , . . .



Geometric Sequence

The nth Term of a Geometric Sequence

DefinitionThe nth term of a geometric sequence has the form

an = a1rn,

where r is the common ratio of consecutive terms of thesequence.



Geometric Sequence

Exercises

Write the first five terms of the geometric sequence.

1. a1 = 6, r = 2

2. a1 = 1, r = 13

3. a1 = 6, r = −14

4. a1 = 3, r =√

5

5. a1 = 5, r = 2x

Write an expression for the nth term of the geometric sequence.Then find the indicated term.

1. a1 = 5, r = 32 , n = 8

2. a1 = 64, r = −14 , n = 10

3. a1 = 1, r =√

3, n = 8

4. a1 = 1000, r = 1.005, n = 60



Geometric Sequence

Summation Notation

DefinitionThe sum of first n terms of a sequence is represented by

n∑

i=1

ai = a1 + a2 + a3 + a4 + · · · + an ,

where i is called the index of summation, n is the upper limit ofsummation, and 1 is the lower limit of summation.

Properties of Sums

1.∑n

i=1 c = cn, c is a constant.

2.∑n

i=1 cai = c∑n

i=1 ai , c is a constant.

3.∑n

i=1(ai + bi) =∑n

i=1 ai +∑n

i=1 bi

4.∑n

i=1(ai − bi) =∑n

i=1 ai − ∑ni=1 bi



Geometric Sequence

The Sum of a Finite Geometric Sequence

DefinitionThe sum of the finite geometric sequence

a1, a1r , a1r2, a1r3

, a1r4, . . . , a1rn−1

with common ratio r 6= 1 is given by

Sn =∑n

i=1 a1r i−1 = a1

(

1−rn

1−r

)



Geometric Sequence

ExercisesFind the sum of the finite geometric sequence.

1.∑10

n=1

(

52

)n−1

2.∑8

n=1 5(

−32

)n−1

3.∑10

i=1 2(

14

)i−1

4.∑12

i=1 16(

12

)i−1

5.∑40

n=0 5(

35

)n

6.∑20

n=0 10(

15

)n

7.∑6

n=0 500 (1.04)n

8.∑50

n=0 10(

23

)n−1

9.∑25

i=0 8(

−12

)i

10.∑100

i=0 15(

23

)i−1



Geometric Series

Geometric Series

The summation of the terms of an infinite geometric sequence iscalled an infinite geometric sequence or simply a geometricsequence.

DefinitionIf |r | < 1, the infinite geometric series

a1 + a1r + a1r2 + a1r3 + · · · + a1rn−1 + · · ·

has the sum

S =∞

∑

i=0

a1r i =a1

1 − r.



Geometric Series

Exercises

Find the sum of the infinite geometric sequence.

1.∑∞

n=0 2(

23

)n

2.∑∞

n=0 2(

−23

)n

3.∑∞

n=0

(

110

)n

4.∑∞

n=0 4 (0.2)n

5.∑∞

n=0 −10 (0.2)n

6. 9 + 6 + 4 + 83 + · · ·

7. −12536 + 25

6 − 5 + 6 − · · ·


References

References

1. M.J. Acelajado, Y.B. Beronque and F.F. Co. Algebra:

Concepts and Processes. National Bookstore, ThirdEdition, 2005.

2. M. Dugopolski. Algebra for College Students. McGraw-HillPublishing Company, Fourth Edition, 2006.

3. G. Fuller. College Algebra. Litton Educational Publishing,Inc., Fourth Edition, 1977.

4. P.K. Rees, F.W. Sparks and C.S. Rees. College Algebra.McGraw-Hill Publishing Company, Tenth Edition, 1990.

5. E.P. Vance. Modern College Algebra. Addison-WesleyPublishing Company, Inc., Third Edition, 1975.

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