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CHAPTER 7
Exponential and Logarithmic
Functions
Ch 7.1 Exponential Growth and Decay
Population Growth
In laboratory experiment the researchers establish a
colony of 100 bacteria and monitor its growth. The
experimenters discover that the colony triples in
population everyday
t
P(t)
Population P(t), of bacteria in t days
P(0) = 100
0
100
P(1) = 100.3
1
300
P(2) = [100.3].3
2
900
P(3) =
3
2700
P(4) =
4
8100
P(5) =
5
24,300
The function P(t) = 100(3) t
The no. of bacteria present after 8 days= 100(3) 8 = 656, 100
After 36 hours bacteria present 100 (3)1.5= 520 (approx)
Graph
25,000
Graph Of Exponential Growth ( in Graph)
20,000
15,000
10,000
5000
Population
1
2
3
4
5
Days
Growth or Decay Factors
Functions 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(
1
2
)t
, Decreasing, 1
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
is a decay factor
Comparing Linear Growth and Exponential
Growth (pg 426)
Linear Function
Let consider the two functions
Exponential function
E(t) = 5.2 t
L(t) = 5 + 2t and
L(t) or E(t)
E(t) = 5.2
50
L(t) = 5 + 2t
0
1
2
3
4
5
t
L(t)
E(t)
0
5
5
1
7
10
2
9
20
3
11
40
4
13
80
t
t
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 + b
Slope = m = 10  4
30
L(t) = 2t + 4
t
0
3
6
9
12
L(t)
4
10
16
22
28
b) Flue grows exponentially
E(t) = E0 at
E0 = 4,
E(t) = 4 at
10 = 4 at
10 = t,
a
4
10
4
t
0
3
6
9
12
L(t)
4
10
25
62
156
= a3 , t = 3
10
=
4
5
 
2
a=
E(t) = 4(1.357)t
3
1
3
= 1.357
Graph
Flu cases grow linearly
Flu grows exponentially
7.2 Exponential Functions ( Pg 434)
We define an exponential function to be one of the form
f(x) = abx , where b > 0 and b = 1, a = 0
If 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 number
If 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 = a
For examples , we find y-intercepts are
f(0)= 50 = 1
P(0) = 250(1.7) 0 = 250
G(0) = 2.4(0.3) 0 = 2.4
The 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
f(x)= 2
x
f(x)
-3
1/8
-2
1/4
-1
1/2
0
1
1
2
2
4
3
8
g(x)= (1/2)x
x
(3, 8)
x
g(x)
-3
8
-2
4
-1
2
0
1
1
1/2
2
1/4
3
1/8
(-3, 8)
( 0,2)
(-2, 1/4)
(-3, 1/8)
-5
( 0,1)
( 0,1)
(2, 1/4)
(3, 1/8)
5
-5
5
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
PRESS Y=
STAT, RIGHT, 0, FOR EXP REG, PRESS ENTER
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) = 800
According to statement 800 = 50.2 x
Dividing both sides by 50 yields
16 = 2x
What power must we raise 2 in order to get 16 ?
Because 2 4 = 16
Log 16 = 4
2
In 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 Equations
Pg( 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)
=13.51inch
Check in gr. calculator
7.4 Logarithmic Functions (pg 461- 462)
Logarithmic function
Inverse of function
x
f(x) =x
-2
-8
-1
-1
- 1/2
-1/8
0
1/2
0
1/8
3
x
g(x)=
-8
-2
-1
-1/8
0
-1
-1/2
0
1/8
1/2
1
1
1
1
2
8
8
2
3
3x
x
f(x) =2 x
-2
1/4
-1
1/2
0
1
1
2
x
g(x) = log 2 x
-1/4
-2
1/2
-1
1
0
2
2
1
4
4
2
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 Functions
Use Log key on a calculator
Ex 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
In calculator
• C) 2f(16) + 1 = 2 log 10 16 + 1
• = 2(1.204) + 1 = 3.408
Example 2, pg 464
Evaluate the expression
log 10
T=
Mf + 1
Mo
K
For k = 0.028, Mf = 1832 and Mo = 15.3
T = log 10 1832 + 1
15.3
0.028
In calculator
= log 10 ( 120.739)
0.028 =
2.082
0.028
= 74.35
Ex 7.4 ,No 12, Pg 469
T= H
log 10
, H= 5730, N = 180, N0= 920
N
N0
log 10 1
2
In calulator
T = 5730 log 10 180
920
log 10 (
= 13486.33975
1
2
)
7.6 The Natural Base ( pg 484)
• Natural logarithmic function (ln x)
In general, y= ln x if and only if ey = x
y=ex
• 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=x
y = ln x
Properties of Natural Logarithms (pg 485)
If x, y > 0, then
1.
ln(xy) = ln x + ln y
2.
ln x = ln x – ln y
y
3.
ln xm = m ln x
Useful Properties
ln ex = x e lnx = 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)
t
0
5
10
15
20
25
30
N(t)
6000
7328
8951
10,933
13,353
16,310
19,921
15000
c) graph
d) After 24 years, there were N(24) = 6000 e 0.04 ( 24) = 15,670
Let N(t) = 100,000;
100,000 = 6000 e 0.04t
DIVIDE BY 6000 AND REDUCE
 50  = e 0.04 t
 
10000
e)
5000
 3 
Change to logarithmic form :
0.04t = loge  50 
 
 3 
1
t = 0.04
 50 

 3 
ln 
= 70.3
50
= ln  
 3 
( divide by 0.04)
There will be 100,000 bacteria present after about 70.3
10
20
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 = ln 2.7 = 0.8277
1 .2
Solve each equation for the specified variable
No. 31 y = k(1- e - t), for t
y = 1- e – t (Divide by k)
y
k
e –t =1– k
-t = ln( 1- y )
k
t = - ln ( 1 -
y)
k
= ln
 k 


k y