Section 3.5

Exponential and Logarithmic Models

Compound InterestThe compound interest formula:

A is the amount in the account after t years.

P is the principal.

r is the annual interest rate.

n is the number of pay periods per year.

1

ntr

A Pn

Example 1An investment of $5,000 is made into an account that pays 6% annually for 10 years. Find the amount in the account if the interest is compounded:

a) annually

b) quarterly

c) monthly

d) daily

105000 1 .06A $8,954.24

4 10.06

5000 14

A

$9,070.09

12 10.06

5000 112

A

$9,096.98

365 10.06

5000 1365

A

$9,110.14

a.

b.

c.

d.

Example 2If interest is compounded semiannually, find how long it will take for $2500 invested at 10% per year to double.

P =

r =

A =

n =

$2500

0.1

$5000

2

20.1

5000 2500 12

t

22 1.05

t

2ln 2 ln 1.05

t

ln 2 2 ln1.05tln 2

2ln1.05

t

2 14.2067t 7.103t

It will take a little more than 7 years.

Compound ContinuouslyAfter t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by

A = Pert

Example 3Use the principal, annual interest rate, and years from example 1 but compound continuously. Find the amount in the account.

.06 105000A e$9,110.59

Example 4If after 15 years an investment of $100 compounded continuously is worth $300, what is annual interest rate?

300 = 100e15r

3 = e15r

ln 3 = ln e15r

ln 3 = 15r

r ≈ 0.0732

The interest rate is about 7.32%.

Exponential Growth and Decay Formulas

1. Half-Life Formula

where

Nt = the quantity that still remains and has not yet decayed after a time t,

N0 = the initial quantity of the substance that will decay,

k = the half-life of the decaying quantity.

0

1

2

t

k

tN N

2. Exponential Decay Model

where b > 0

bxy ae

3. Exponential Growth Model

where b > 0

bxy ae

Example 5Suppose the half-life of a certain radioactive substance is 20 days and there are 5 grams present initially. How much will be left in 70 days? Round to nearest hundredth of a gram.

t =

N0 =

k =

70 days

5 grams

20 days

70

205 .5N

0.44gramsN

Example 6For a certain strain of bacteria the growth constant (k) = 0.658 when t is measured in hours. How long will 15 bacteria take to increase to 250 bacteria?

N0 =

Nt =

250 = 15e0.658t

15

250

0.658250 15 te

0.658250

15te

0.658250ln ln

15te

2.81341 0.658t4.2757t

It will take more than 4 hours to increase the bacteria to 250 bacteria.

Example 7According to the U.S. Bureau of the Census in 1990 there were 22.4 million residents of Hispanic origin living in the United States. By 2000, the number had increased to 35.3 million. a. Write the exponential growth equation.

0kt

tN N e 1035.3 22.4 ke

101.57589 keln1.57589 10k

0.046k

0.04622.4 ttN e

b. Use the model to project the Hispanic population in 2010.

In 2010 the Hispanic resident population will be about 56.2 million people.

0.04622.4 ttN e

0.046 2022.4tN e56.2081tN

c. In what year will the Hispanic resident population reach 60 million?

The Hispanic resident population will reach 60 million people sometime in the year 2011.

0.04660 22.4 te0.0462.67857 te

ln 2.67857 0.046t21.4t

Gaussian Model

The Gaussian Model is

This type of model is commonly used in probability and statistics to represent populations that are normally distributed. The graph of a Gaussian model is called a bell-shaped curve.

2

.x b

cy ae

For standard normal distributions, the model takes the form

The average value for a population can be found from the bell-shaped curve by observing where the maximum y-value of the function occurs. The x-value corresponding to the maximum y-value of the function represents the average value of the independent variable.

2

21

2

x

y eπ

Example 8Last year, the math scores for students in a particular math class roughly followed the normal distribution given by

where x is the math score. Sketch the graph of this function and use it to estimate the average math score.

274

1140.0399 , 30 110x

y e x

x

y

0.025

0.05

x

y

(74, 0.04)

0.025

0.05

The average score for students in the math class was 74.

Logistic Growth ModelSome populations initially have rapid growth, followed by a declining rate of growth.

One model for describing this type of growth pattern is the logistic curve given by the function

Where y is the population size and x is the time.

,1 rx

ay

be

An example is a bacteria culture that is initially allowed to grow under ideal conditions, and then under less favorable conditions that inhibit growth.

A logistic growth curve is also called a sigmoidal curve.

Example 9On a college campus of 7500 students, one student returns from vacation with a contagious disease and long-lasting virus. The spread of the virus is modeled by

where y is the total number of students affected after t days.

0.9

7500, 0

1 7499 ty t

e

The college will cancel classes when 30% or more of the students are infected.

a. How many students will be infected after 4 days?

Approximately 36 students will be infected.

0.9 4

7500

1 7499y

e

b. After how many days will the college cancel classes?

0.9

75000.3 7500

1 7499 te

0.9 75001 7499

2250te

0.97499 2.333333te 0.9 0.000311te

0.9 ln 0.000311t 8.97t

Classes will be canceled after 9 days.

Logarithmic Modelln , or logy a b x y a b x

Example 10On the Richter scale, the magnitude R of an earthquake of intensity I is given by

where I0 = 1 is the minimum intensity used for comparison. Find the magnitude of R of an earthquake of intensity 68,400,000 to the nearest hundredth.

0

logI

RI

I = 68,4000

68400000log

17.84

R

R