1. Exponential and Logarithmic Functions
Prof. M. Alonso pH

2. OBJETIVES Define exponential and logarithmic functions
Graph both functions
Solve exponential and logarithmic equations
Apply the properties of logarithms
Solve problems

3. Note the variable is an exponent
Exponential Function
Definition An exponential function is 
f(x) = Abx
 where b > 0 and b  1. A is a real number
The domain of this function are the real numbers .
Note the variable is an exponent

4. EXAMPLeS
F(x) = 2x H(x) = 102x+3 G(x) = (¼)x-1 Q(x) = (½)2x P(x) = (3) –x M(x) = (¾)4x

5. NATURAL Base e
A common base used in the sciences is the base e also called natural base. e is a symbol such as π which represents the number e =
Thus we have the exponential function

6. Base e
To find the value of e , use the ex button (generally you’ll need to hit the 2nd function or blue button first to get it depending on the calculator). After hitting the ex, you then enter the exponent 1 and push = or enter.
If you have a scientific calculator that doesn’t graph you may have to enter the 1 before hitting the ex button . You should get

7. GrAph of f(x) = 2x Remember! f(-3)= 2-3=.125
Let us plot some points. X Y
Remember!
f(-3)= 2-3=.125
A negative exponent means:

8. Characteristics about F(X) = 2x
The domain is the real numbers
The range is (0, ∞ )
Y - Intercept f(0) = 20 = 1 , (0, 1)
X - Intercepto 2x = 0 There are no x intercepts because there is no x value that you can put in the function to make it = 0.
The graph is always increasing

9. Graph f(x)= x y -3 8 -2 4 -1 2 1 .5
.25 3
.125

10. Characteristics about
Domain- real numbers
Range – (0, ∞)
Y- intercept – (1, 0)
X – intercept - There are no x intercepts
The graph is decreasing

11. summary If b > 1 then the graph of f(x) = bx is increasing..
The graphs can be increasing or decreasing. By looking at the function we know if the graph is increasing or decreasing.
If b > 1 then the graph of f(x) = bx is increasing..
If 0 < b < 1, then the graph of f(x) = bx ss decreasing.

12. Exponential Equations
An exponential equation is one in which a variable occurs in the exponent:
3x = 9
52x+1 = 125x
6x = 7
42x-2 = 8
5 = 25x-2

13. How to solve exponenTial Equations
 Theorem: If bx = by then x = y.
This says that if both sides of the equation have the same base, then the exponents are equal

14. Remember, if the bases are equal then the exponents are equal.
Example: solve 3x = 9
Solution: Use the theorem
The left hand side is 3 to the something. Re-write the right hand side, 9, as 3 to the power 2
3x = 9
3x = 32
Thus, x = 2
Remember, if the bases are equal then the exponents are equal.

15. Example: Solve 52x+ 1 = 125x 52x+ 1 = 125x 52x+1 = (53)x
Thus, 2x+1 = 3x
1 = 3x -2x
1 = x
Re-write 125 as 53.

16. Example: solve 42x-2 = 8
42x-2 = 8 (22)2x-2 = 23 24x-4 = 23 Thus: 4x -4 = 3 4x = 7 x = 7/4

17. Example:solve 51/2 = (52) x-2 51/2 = 5 2x-4 Thus: ½ = 2x - 4
51/2 = (52) x-2
51/2 = 5 2x-4
  Thus: ½ = 2x - 4
½ + 4 = 2x
9/2 = 2x
9/4 = x

18. Example: solve 6x = 7
With the procedure used in the previous cases we can not solve this equation since there is no way to put the bases equal.
Later we will use another technique to solve this equation.

19. Applications
Exponential functions have many applications, both in the Natural Sciences and in Business Administration. We will see some interesting applications.

20. Compound interest
Compound interest (or compounding interest) is interest calculated on the initial principal and also on the accumulated interest of previous periods of a deposit.

21. Compound interest
If one invests a certain amount of money P into a savings account that pays an interest rate r and the interest is calculated in certain periods, then after t years the person gets the following amount A of money:
A = P( 1 + r/n)nt  

22. Compound interest
The letters in the formula A = P( 1 +r/n)nt represent:
P the principal, the money that you deposit
r the annual rate of interest
n the number of times that interest is compounded per year ( if monthly, then n = 12; if quarterly then n = 4.
t number of years the amount is deposited
A amount of money accumulated after t years,

23. A = P( 1 +r/n)nt
The formula is an exponential function because the variable t is in the exponent.
That is, the amount of money A that one will have in the future depends on the time that the principal is in the account.

24. EXample
If you invest $ 2000 in a 6.5% IRA compounded monthly, after 5 years how much money does the person have?
= 2000 ( )60 
= 2000 ( )60
  =
Use a calculator

25. Logarithmic Function
Definition: The logarithmic function is defined as: f(x) = logbx where b > 0, b  1. The domain is the interval (0,  ).

26. Examples f(x) = log3 (x + 1) g(x) = log x2 h(x) = log2( x -3 )
f(x) = log3 (x + 1)
g(x) = log x2
h(x) = log2( x -3 )
p(x) = log½ (3x3 + 2x - 4)

27. Logarithms
There are two bases that are used a lot when working with logarithms: base 10 and base e. When we refer to base 10 we only write log x, that is, we do not indicate the base because it is understood that the base is 10.
When we refer to the base e, called natural base, we write ln x; That is, ln x means logex.
Both log x and ln x are keys that appear in any scientific calculators.

28. Logarithms
You should check how your calculator works.
If we want to find log 100 we just have to press the log key and then 100. The answer is 2
To find ln 3, press the ln button and then 3 . The answer is ln 3 =

29. Logarithms What is a logarithm?
Find this logarithms of:
Log1000
Log 100
Log 10
Log 1
Log .01
Log .001
What is a logarithm?
a quantity representing the power to which a fixed number (the base) must be raised to produce a given number.

30. Answer Logarithm base 10 value Note Log 1000 3 103=1000 Log 100 2
102=100
Log 10 1
101=10
Log 1
100=1
Log .01 -2
10-2=.01
Log .001 -3
10-3=.001

31. Logarithm
It follows that a logarithm is an exponent. That is, a logarithm is the exponent to which the base must be raised to obtain the number. Therefore, the equivalent form is
Exponential form
Logarithm form

32. Summary y = log2 x 2y = x y = log2 x is equivalent to 2y = x .
EXPONENT
y = log2 x y = x
BASE
NUMBER

33. Remember what a logarithm is!!
GrAph of g(x) = log2x x y
Log2(¼)=-2
Log2 (½)=-1 1
Log21 = 0 2
Log22= 1 4
Log24= 2 8
Log28 = 3
Remember what a logarithm is!!
Powers of 2

34. Graph of g(x) = log2x
Domain is the interval
( 0,  ).

35. Compare F(x) = 2x g(x) = log2x g(x) = log2x F(x) = 2x
Compare these two graphs and their value tables, what do you observe?
g(x) = log2x
F(x) = 2x

36. logarithms Thus, log 2 ( 1/64) = -6 Find log2 (1/64) Solution:
Write log2 (1/64) = N
Change to an equivalent form: 2N = 1/64
Solve this exponential equation
N = -6
Thus, log 2 ( 1/64) = -6

37. Previous example 2N = 1/64 We have an exponential equation
Thus, N = -6. The answer is:

38. Finding logaritHms
Find log381
Thus, N = 4. The answer is log381 = 4

39. The Five Basic Properties of Logs
Let A, C and N be real numbers
1. logb 1 = 0
2. logbb = 1
3. logb (AC) = logb A + logb C
4. logb (A/C) = logb A - logb C
5. log b AN = N logb A

40. logarithm Properties Simplify logarithms Solve exponential equations
Solve logarithmic equations

41. EXample: log3 x + log3 x2 Simplify log3x + log3x2
We use property #3.
Simplify log3x + log3x2
log3 x + log3 x2 = log3 (x)(x2 ) = log3 x3

42. Use logarithm properties!!!!
Example:
If logb2 = .23 and logb5 = Find
Logb10
Logb8
Logb ½
Logb25b
Use logarithm properties!!!!

43. Answers: Logb10 = Logb(2)(5) = Logb2 +Logb5 = .23 + .42 = .65
Write 10 as a product
Logb10 = Logb(2)(5)
= Logb2 +Logb5 =
= .65
Usa property #3
Substitute
Add

44. Answers Logb8 = Logb23 = 3Logb2 = 3(.23) = .69 Write 8 as a power of 2
Use property #5
Substitute
Multiply

45. Answers Logb(1/2) = Logb1 - Logb2 = 0 - .23 = -.23 Use property #4=
= -.23
Use property #4
Substitute
Add

46. Answers
Use property #3
Logb25b = Logb25 + logbb = Logb = 2Logb5 + 1 = 2(.42) + 1 = 1.84
Use #5 and #2
Substitute
Add

47. Solving logarithmic equations
Solve log 3 (x - 2) + log 3 (x - 4) = 2

48. Change to exponential form Use quadratic formula to solve
Solution
Use property # 3
log 3 (x - 2) + log 3 (x - 4) = 2
log 3 (x - 2)(x - 4) = 2
log 3 (x2 –6x + 8) = 2
(x2 –6x + 8) = 32
x2 –6x + 8 – 9 = 0
x2 –6x –1 = 0
Multiply
Change to exponential form
Quadratic
equation
Use quadratic formula to solve

49. Quadratic formula
The only solution is

50. exponenTial equations
Solve 2x = 5
Since 5 cannot be written as a power of 2, we need another procedure to solve the equation.

51. Exponential equations
To solve an exponential equation, take the log of both sides ( base 10 or base e) and solve for the variable.
2x = 5
log 2x = log 5

52. Example Solve 2x = 5 log 2x = log 5 x log 2 = log 5 Use property # 5
Solve for x
Use the calculator

53. EXAmplE
Solve
Use property # 5
Common Factor x

54. Applications
Exponential functions are used to model carbon date artifacts, population growth, exponential decay, and compound interest.

55. Applications
Let’s look at a general form for population models. Most of the time, we start with an equation that looks like
P0 is the initial population
k is the growth rate
t time

56. Example
Suppose that the population of a certain country grows at an annual rate of 3%. If the
current population is 5 million, what will the population be in 10 years?
Solution:
P(t) = 5e0.03(10) = … million

57. Example
In the same country as in the previous example, how long will it take the population to reach 10 million?
Solution:
10 = 5e0.03t
10/5 = e0.03t
2 = e0.03t
ln 2 = ln e0.03t
isolate the exponential
apply the natural log function

58. Example ln 2 = ln e0.03t Ln 2 =.03t lne Ln2 = .03t Ln2/.03 = t
23.10 = t , thus 23 years
Property #5 and #2

59. END