1. Exponential and Logarithmic Functions
CHAPTER 7
Exponential and Logarithmic Functions

2. 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
Population P(t), of bacteria in t days
P(0) = 100
P(1) = 100.3
P(2) = [100.3].3
P(3) =
P(4) =
P(5) =
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 (3)1.5= 520 (approx) t
P(t)
100 1
300 2
900 3
2700 4
8100 5
24,300
Graph

3. Graph Of Exponential Growth ( in Graph)
25,000
20,000
15,000
10,000
5000
Graph Of Exponential Growth ( in Graph)
Population
Days

4. 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( ) t , Decreasing, is a decay factor
For bacteria population we have
P(t) = t
Po = 100 and a = 3
Percent Increase Formula
A(t) = P(1 + r) t

5. Comparing Linear Growth and Exponential Growth (pg 426)
Linear Function Exponential function
Let consider the two functions L(t) = 5 + 2t and E(t) = 5.2 t
L(t) or E(t) t
L(t)
E(t) 5 1 7 10 2 9 20 3 11 40 4 13 80
E(t) = 5.2 t 50
L(t) = 5 + 2t t

6. 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

7. Graph Table

8. 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 =
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 3 6 9 12
L(t) 4 10 16 22 28 t 3 6 9 12
L(t) 4 10 25 62
156
Graph

9. Flu grows exponentially
Flu cases grow linearly

10. 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

11. 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

12. Graphs of Exponential Functions
g(x)= (1/2)x
f(x)= 2x x
g(x) -3 8 -2 4 -1 2 1
1/2
1/4 3
1/8 x
f(x) -3
1/8 -2
1/4 -1
1/2 1 2 4 3 8
(3, 8)
(-3, 8)
( 0,2)
(-2, 1/4)
(2, 1/4)
( 0,1)
( 0,1)
(-3, 1/8)
(3, 1/8)

13. Using Graphing Calculator Pg 437
y = 2x y = 2x y = 2x+3

14. Graphical solution of Exponential Equations by Graphing Calculator ( Ex- 5, Pg –440)
Enter y1 and y Zoom Trace

15. Exponential Regression (Pg 441)
STAT ENTER STAT, RIGHT, 0, FOR EXP REG, PRESS ENTER
PRESS Y= VARS, 5, RIGHT, RIGHT, ENTER PRESS ZOOM 9

16. 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
Log2 16 = 4
In other words, we solve an exponential equation by computing a logarithm.
Check x = 4
P(4) = x = 800

17. 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= because b0 = 1
log b b x = x because bx = b x

18. Steps for Solving Exponential Equations Pg( 454)
Isolate the power on one side of the equation
Rewrite the equation in logarithmic form
Use a calculator, if necessary, to evaluate the logarithm
Solve for the variable

19. 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 = Ymin= 30
A= 20,320 feet= 20,320(1/5280) = miles ( 1mile = 5280 feet)
P = 30(10) –(0.09)(3.8485)
=13.51inch
Check in gr. calculator

20. 7.4 Logarithmic Functions (pg 461- 462)
Inverse of function
Logarithmic function x
f(x) =x 3 -2 -8 -1
- 1/2
-1/8
1/2
1/8 1 2 8 x
g(x)=
- 8 -2 -1
-1/8
-1/2
1/8
1/2 1 8 2 x
f(x) =2 x -2
1/4 -1
1/2 1 2 4 x
g(x) = log 2 x
-1/4 -2
1/2 -1 1 2 4

21. 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

22. Evaluating Logarithmic Functions Use Log key on a calculator Ex 7
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 = 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
= 2(1.204) + 1 = 3.408
In calculator

23. 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
= log 10 ( ) = 74.35 =
In calculator

24. Ex 7.4 ,No 12, Pg 469 T = H log 10 , H= 5730, N = 180, N0= 920 log 10
T = log =
920
log 10 ( ) N0
In calulator

25. 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 = x
y = ln x

26. Properties of Natural Logarithms (pg 485)
If x, y > 0, then
ln(xy) = ln x + ln y
ln = ln x – ln y
ln xm = m ln x
Useful Properties
ln ex = x e lnx = x

27. 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.
Growth law N(t) = 6000 e 0.04t
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
= e 0.04 t
Change to logarithmic form :
0.04t = loge = ln
t = ln = ( divide by 0.04)
There will be 100,000 bacteria present after about 70.3 t 5 10 15 20 25 30
N(t)
6000
7328
8951
10,933
13,353
16,310
19,921
15000
10000
5000

28. Ex 7.6, Pg 492 Solve, Round your answer to two decimal places
No = 5.3 e 0.4x
2.7 = e 1.2x ( Divide by 2.3 )
Change to logarithmic form
1.2x = ln 2.7
x = =
Solve each equation for the specified variable
No y = k(1- e - t), for t
= 1- e – t (Divide by k)
e – t = 1 –
-t = ln( )
t = - ln ( ) = ln