calculus for business, economics, and the social and life ...d. for continuously compounded interest...

Chapter 4Exponential and

Logarithmic Functions

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What We Will Learn?

In this Chapter, we will encounter some important concepts:

Exponential Functions

Logarithmic Functions

Differentiation of Logarithmic and

Exponential Functions

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4.1 Exponential Functions

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Exponential Functions

If b is a positive number other than 1 (b>0, b≠1),

there is a unique function called the exponential

function with base b that is defined by

f(x)=bx for every real number x

Such function can be used to describe exponential and

logistic growth and a variety of other important

quantities.

Definition of for Rational Values of n (and b>0)

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Integer powers: If n is a positive integer,

Fractional powers: If n and m are positive integers,

where denotes the positive mth root.

Negative powers:

Zero power:

nb

facters n

n bbbb

m nn

mmn bbb /

m b

n

n

bb

1

10 b

Example

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Figure below shows graphs of various members of the

family of exponential functions xby

NOTE: Students often confuse the power function

with the exponential function

bxxp )(

xbxf )(

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The natural exponential function

The natural exponential function is

where

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n 10 100 1000 10,000 100,000

2.59374 2.70481 2.71692 2.711815 2.71827

Continuous Compounding of Interest

If P is the initial investment (the principal) and r is the interest rate (expressed as a decimal), the balance B after the interest is added will be

B=P+P r=P(1+r) dollars

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Example

Suppose $1,000 is invested at an annual interest rate of 6%. Compute the balance after 10 years if the interest is compounded

a. Quarterly b. Monthly c. Daily d. Continuously

a.To compute the balance after 10 years if the interest is compounded quarterly, use the formula

with t=10, p=1,000, r=0.06, and k=4.

15

kt

k

rptB

1)(

02.814,1$4

06.01000,1)10(

40

B

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b. This time, take t=10, p=1,000, r=0.06, and k=12 to get

40.819,1$12

06.01000,1)10(

120

B

c. Take t=10, p=1,000, r=0.06, and k=365 to obtain

03.822,1$365

06.01000,1)10(

650,3

B

d. For continuously compounded interest use the formula rtpetB )(

with t=10, p=1,000, and r=0.06.

12.822,1$000,1)10( 6.0 eB

This value, $1,822.12, is an upper bound for the possible balance.

No matter how often interest is compounded, $1,000 invested at

an annual interest rate of 6% can not grow to more than

$1,822.12 in 10 years.

Present Value & Future Value

Present value is the interest accumulation process in reverse. Rather than adding interest to a principal to determine a sum, it is in effect subtracted from a sum to determine a principal.

The present value P of future value F with interest rate r per conversion period for n period is given by the formula

is the present value interest factor (PVIFn) (or discount factor) of n periods.

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If the interest rate is i*100% per year and the interest is

compounded continuously for t years, then the present

value P for a given amount of future value F can be

calculated from

where exp(−it) is the continuous PVIF.

i = annualized interest rate,

t = number of years

m = compound periods per year

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Present value of $1.00 when interest (i=5%) is compounded

quarterly for a year

Example

Sue is about to enter college. When she graduates 4 years from now, she wants to take a trip to Europe that she estimates will cost $5,000. How much should she invest now at 7% to have enough for the trip if interest is compounded:

a. Quarterly b. Continuously

The required future value is F=$5,000 in t=4 years with r=0.07.

a. If the compounding is quarterly, then k=4 and the present value is

b. For continuous compounding, the present value is

Thus, Sue would have to invest about $9 more if interest is compounded quarterly than if the compounding is continuous.

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08.788,3$4

07.01000,5

)4(4

P

92.778,3$000,5 )4(07.0 eP

More Example

Suppose that you plan to buy a luxury car four years from now and that car will cost $55,000 at that time. If your bank’s savings deposit interest rate is 5% per year, what is the amount that you have to deposit now (present value) in your savings account in order to grow enough interest and with the principal to buy your dream car in the future? Assume the interest is compounded: (a) annually; (b) quarterly; (c) weekly; or (d) continuously, for a year. (e) Show your results in a table of PV calculation with separate FV, PVIF, and PV columns.

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a) annually P = $55,000(1 + 0.05)−4 = $45,248.64;b) quarterly P = $55,000(1 + 0.05/4)−4∗4 = $45,086.05;c) weekly P = $55,000(1 + 0.05/52)−52∗4 = $45,034.52;d) continuously P = $55, 000e−0.05∗4 = $45,030.19;e) .

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4.2 Logarithmic Functions

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Logarithmic Functions

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Example

Use logarithm rules to rewrite each of the following expressions in terms of .

a. b. c.

a.

b.

c.

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3log and 2log 55

3

5log5

8log536log 5

15log since 3log1

rulequotient 3log5log3

5log

55

555

rulepower 2log32log8log 5

3

55

rulepower 3log222log

ruleproduct 3log2log)32(log36log

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2

5

2

5

22

55

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Doubling Time

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4.3 Differentiation of Logarithmic and Exponential Function

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Limit and Derivative

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Example

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Example

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Exponential Growth and Decay

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Exponential Growth and Decay

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Example

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Logarithmic Differentiation

Taking the derivatives of some complicated functions can be simplified by using logarithms.

Example

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More Examples

Differentiate each of these function

a. b.

a. To differentiate , we use logarithmic differentiation as follows:

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xy 2 xy 3log

xy 2

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b.

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Optimal Holding Time

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The relative rate of change of a quantity Q(x) can be

computed by finding the derivative of ln Q.

'( )(ln )

( )

d Q xQ

dx Q x

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Summary

Exponential Functions, Basic Properties of Exponential

Functions, The Natural Exponential Base e.

Compound Interest, Continuously Compounded Interest,

Present Value.

Exponential Growth and Decay.

Logarithmic Functions, The Natural Logarithm.

Differentiation of Logarithmic and Exponential

Functions.

Optimal Holding Time.