CHAPTER 7

Exponential and Logarithmic Functions

Ch 7.1 Exponential Growth and Decay

Population GrowthIn laboratory experiment the researchers establish acolony of 100 bacteria and monitor its growth. Theexperimenters discover that the colony triples inpopulation everyday

Population P(t), of bacteria in t daysP(0) = 100P(1) = 100.3P(2) = [100.3].3P(3) =P(4) =P(5) = The function P(t) = 100(3) t

The no. of bacteria present after 8 days= 100(3) 8 = 656, 100After 36 hours bacteria present 100 (3)1.5= 520 (approx)

t P(t)

0 100

1 300

2 900

3 2700

4 8100

5 24,300

Graph

Graph Of Exponential Growth ( in Graph) 25,000

20,000

15,000

10,000

5000

1 2 3 4 5 Days

Population

Growth or Decay FactorsFunctions that describe exponential growth or decay can be expressed in the standard form

P(t) = Po a t , where Po = P(0) is the initial value of the function and a is the growth or decay factor.

If a> 1, P(t) is increasing, and a = 1 + r, where r represents percent increase Example P(t) = 100(2)t Increasing 2 is a growth factor

If 0< a < 1, P(t) is decreasing, and a = 1 – r, where r represents percent decrease

Example P(t) = 100( ) t , Decreasing, is a decay factor

For bacteria population we have P(t) = 100.3 t

Po = 100 and a = 3

Percent Increase Formula

A(t) = P(1 + r) t

2

1

2

1

Comparing Linear Growth and Exponential Growth (pg 426)

Let consider the two functions L(t) = 5 + 2t and E(t) = 5.2 t

t L(t) E(t)

0 5 5

1 7 10

2 9 20

3 11 40

4 13 80L(t) = 5 + 2t

E(t) = 5.2 t

0 1 2 3 4 5 t

50

L(t) or E(t)

Linear Function Exponential function

Ex 7.1, Pg 429

No 2. A population of 24 fruit flies triples every month. How many fruit flies will there be after 6 months? After 3 weeks? ( Assume that a month = 4 weeks)

• P(t) = P0 at

1st part P(t) = 24(3)t ,

P0= 24, a = 3, t = 6 months

P(6) = 24 (3)6= 17496

2nd part t = 3 weeks = ¾ th months

P(3/4) = 24(3) ¾ = 54.78= 55 (approx)

Graph and table

Graph Table

No 42. Over the week end the Midland Infirmary identifies four cases of Asian flu. Three days later it has treated a total of ten cases

a) Flu cases grow linearly L(t) = mt + bSlope = m =

L(t) = 2t + 4

b) Flue grows exponentially

E(t) = E0 at

E0 = 4, E(t) = 4 at 10 = 4 at

= at,

= a 3 , t = 3

a = = = 1.357

E(t) = 4(1.357)t

t 0 3 6 9 12

L(t) 4 10 16 22 2803

410

t 0 3 6 9 12

L(t) 4 10 25 62 1564

10

3

4

10

Graph

4

10

2

53

1

Flu cases grow linearly Flu grows exponentially

7.2 Exponential Functions ( Pg 434)

We define an exponential function to be one of the formf(x) = abx , where b > 0 and b = 1, a = 0If b < 0 , bx will be negative then b is not a real number for some value of x

For example b = -3 , bx = (-3) x , f( ½) = ( -3) ½, is an imaginary numberIf b= 1, f(x) = 1 x = 1 which is constant function

Some examples of exponential functions are f(x) = 5x ,   P(t)= 250(1.7)t

g(t) = 2.4(0.3) t

The constant a is the y-intercept of the graph because

f(0) = a.b0= a.1 = aFor examples , we find y-intercepts are f(0)= 50 = 1P(0) = 250(1.7) 0 = 250G(0) = 2.4(0.3) 0 = 2.4The positive constant b is called the base of the exponential function

Properties of Exponential Functions (pg 435)

f(x) = abx , where b> 0 and b = 1, a = 0

• 1. Domain : All real numbers

• 2. Range: All positive numbers

• 3. If b> 1, the function is increasing, if 0< b < 1, the function is decreasing

Graphs of Exponential Functions

x f(x)

-3 1/8

-2 1/4

-1 1/2

0 1

1 2

2 4

3 8

x g(x)

-3 8

-2 4

-1 2

0 1

1 1/2

2 1/4

3 1/8

- 5 5 - 5 5

f(x)= 2xg(x)= (1/2)x

(-3, 1/8)

(-2, 1/4)

( 0,1)

(3, 8)

( 0,2)

(3, 1/8)

(2, 1/4)( 0,1)

(-3, 8)

Using Graphing Calculator Pg 437

y = 2x y = 2x + 3 y = 2x+3

Graphical solution of Exponential Equations by Graphing Calculator

( Ex- 5, Pg –440)

Enter y1 and y2 Zoom 6 Trace

Exponential Regression (Pg 441)

STAT ENTER STAT, RIGHT, 0, FOR EXP REG, PRESS ENTER

PRESS Y= VARS, 5, RIGHT, RIGHT, ENTER PRESS ZOOM 9

7.3 Logarithms (Pg 449)

Suppose a colony of bacteria doubles in size everyday. If the colony starts with 50 bacteria, how long will it be before there are 800 bacteria ?

Example P(x) = 50. 2x ,when P(x) = 800According to statement 800 = 50.2

x

Dividing both sides by 50 yields16 = 2x

What power must we raise 2 in order to get 16 ?Because 2 4 = 16Log

2 16 = 4In other words, we solve an exponential equation by computing a logarithm.Check x = 4 P(4) = 50. 2x = 800

Logarithmic Function ( pg 450 - 451)

• y = log b x and x = by

For any base b > 0

• log b b= 1 because b1 = b

• log b 1= 0 because b0 = 1

• log b b x = x because bx = b x

Steps for Solving Exponential EquationsPg( 454)

1. Isolate the power on one side of the equation

2. Rewrite the equation in logarithmic form

3. Use a calculator, if necessary, to evaluate the logarithm

4. Solve for the variable

7.3 No. 40, Pg 458• The elevation of Mount McKinley, the highest mountain in the United

States, is 20,320 feet. What is the atmospheric pressure at the top ?

P(a) = 30(10 )-0.9a , Where a= altitude in miles and

P = atmospheric pressure in inches of mercury

X min = 0 Ymax = 9.4

Xmax = 0 Ymin= 30

A= 20,320 feet= 20,320(1/5280) = 3.8485 miles ( 1mile = 5280 feet)

P = 30(10) –(0.09)(3.8485)

=13.51inchCheck in gr. calculator

7.4 Logarithmic Functions (pg 461- 462)

x f(x) =x 3

-2 -8

-1 -1

- 1/2 -1/8

0 0

1/2 1/8

1 1

2 8

x g(x)=

- 8 -2

-1 -1

-1/8 -1/2

0 0

1/8 1/2

1 1

8 2

Inverse of function

x f(x) =2 x

-2 1/4

-1 1/2

0 1

1 2

2 4

x g(x) = log 2 x

-1/4 -2

1/2 -1

1 0

2 1

4 2

3 3x

Logarithmic function

Properties of Logarithmic Functions (Pg 463)

y = log b x and x = by

1. Domain : All positive real numbers

2. Range : All real numbers

3. The graphs of y = log b x and x = by

are symmetric about the line y = x

Evaluating Logarithmic FunctionsUse Log key on a calculatorEx 7.4, Example 2, pg 464

• Let f(x) = log 10 x , Evaluate the following

• A) f(35) = log 10 35 = 1.544

• B) f(-8) = , -8 is not the domain of f , f(-8),

or log 10 (-8) is undefined

• C) 2f(16) + 1 = 2 log 10 16 + 1

• = 2(1.204) + 1 = 3.408

In calculator

Example 2, pg 464

Evaluate the expression log 10 Mf + 1

T = Mo

K For k = 0.028, Mf = 1832 and Mo = 15.3

T = log 10 1832 + 1

15.3 = log 10 ( 120.739) 2.082 = 74.35

0.028 0.028 = 0.028In calculator

Ex 7.4 ,No 12, Pg 469

T = H log 10 , H= 5730, N = 180, N0= 920

log 10

T = 5730 log 10 180 = 13486.33975

920

log 10 ( )

NN0

2

1

2

1

In calulator

7.6 The Natural Base ( pg 484)

• Natural logarithmic function (ln x)

In general, y= ln x if and only if ey = x

• Example e 2.3 = 10 or ln 10 = 2.3

• In particular

ln e = 1 because e 1 = e

ln 1 = 0 because e0 = 1

y = e x

y = ln xy = x

Properties of Natural Logarithms (pg 485)

If x, y > 0, then

1. ln(xy) = ln x + ln y

2. ln = ln x – ln y

3. ln xm = m ln x

Useful Properties

ln ex = x e lnx = x

y

x

Ex 7.6 (Pg 491)No 9. The number of bacteria in a culture grows according to the function N(t) = N0 e 0.04t , N0 is the number of bacteria present at time t =

0 and t is the time in hours.a) Growth law N(t) = 6000 e 0.04t

b)

c) graph

d) After 24 years, there were N(24) = 6000 e 0.04 ( 24) = 15,670 e) Let N(t) = 100,000; 100,000 = 6000 e 0.04t

DIVIDE BY 6000 AND REDUCE = e 0.04 t

Change to logarithmic form : 0.04t = loge = ln

t = ln = 70.3 ( divide by 0.04)

There will be 100,000 bacteria present after about 70.3

t 0 5 10 15 20 25 30

N(t) 6000 7328 8951 10,933 13,353 16,310 19,921

3

50

04.0

1

3

50

3

50

15000

10000

5000

10 20

3

50

Ex 7.6, Pg 492

Solve, Round your answer to two decimal places

No 22 22.26 = 5.3 e 0.4x

2.7 = e 1.2x ( Divide by 2.3 ) Change to logarithmic form 1.2x = ln 2.7 x = = 0.8277

Solve each equation for the specified variableNo. 31 y = k(1- e - t), for t

= 1- e – t (Divide by k)

e – t = 1 –

-t = ln( 1- )

t = - ln ( 1 - ) = ln

2.1

7.2ln

k

y

yk

k

k

y

k

y

k

y