1. Chapter 4
Exponential and Logarithmic Functions

2. Chapter 4: Exponential and Logarithmic Functions
Chapter Objectives
To introduce exponential functions and their applications.
To introduce logarithmic functions and their graphs.
To study the basic properties of logarithmic functions.
To develop techniques for solving logarithmic and exponential equations.

3. Chapter Outline Exponential Functions 4.1) Logarithmic Functions
Chapter 4: Exponential and Logarithmic Functions
Chapter Outline
Exponential Functions
Logarithmic Functions
Properties of Logarithms
Logarithmic and Exponential Equations
4.1)
4.2)
4.3)
4.4)

4. 4.1 Exponential Functions
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
The function f defined by (for example )
where b > 0, b  1, and the exponent x is any real number, is called an exponential function with base b.

5. Example 1 – Bacteria Growth
The number of bacteria present in a culture after t minutes is given by .
a. How many bacteria are present initially?
b. Approximately how many bacteria are present after 3 minutes?
Solution:
a. When t = 0,
b. When t = 3,
Note that after just 100 minutes, there are
300*(4/3)^100 = e+14 bacteria
The model is only valid for sufficiently small time, after that saturation starts to happen and the growth looks like S-shaped logistic curve

7. Graph the exponential function f(x) = (1/2)x. Solution:
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 3 – Graphing Exponential Functions with 0 < b < 1
Graph the exponential function f(x) = (1/2)x.
Solution:
It is decreasing left to right

8. Properties of Exponential Functions

9. The Natural Base e
In many applications (continuous compounding, radioactive decay, administering drug, etc…), the most convenient choice for a base is this irrational number (need Calculus…)
e  called natural base.
The function given by f (x) = ex is called the natural exponential function.
Be sure you see that for the exponential function
f (x) = ex, e is the constant ,
whereas x is the variable.
Figure 3.9

12. Applications

13. Continuously Compounded Interest.

14. NOTE:
Interest rate per period is called periodic rate.
Interest rate per year is annual percentage rage (APR) also called nominal rate.

15. Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 7 – Population Growth
The population of a town of 10,000 grows at the rate of 2% per year. Find the population three years from now.
Solution:
For t = 3, we have
-same formula works

16. Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 9 – Population Growth
The projected population P of a city is given by where t is the number of years after Predict the population for the year 2010.
Solution:
For t = 20,

18. a. How many milligrams are initially present? Solution: For t = 0, .
Chapter 4: Exponential and Logarithmic Functions
4.1 Exponential Functions
Example 11 – Radioactive Decay
A radioactive element decays such that after t days the number of milligrams present is given by
a. How many milligrams are initially present?
Solution: For t = 0,
b. How many milligrams are present after 10 days?
Solution: For t = 10,

19. HW

20. HW

26. Chapter 4: Exponential and Logarithmic Functions
Example 7 – Finding Half-Life
If a radioactive element has decay constant λ, the half-life of the element is given by
A 10-milligram sample of radioactive polonium 210 (which is denoted 210Po) decays according to the equation. Determine the half-life of 210Po.
Solution:

27. HW:

28. HW:

29. 4.3 Properties of Logarithms

31. a. b. Example 3 – Writing Logarithms in Terms of Simpler Logarithms
Chapter 4: Exponential and Logarithmic Functions
4.3 Properties of Logarithms
Example 3 – Writing Logarithms in Terms of Simpler Logarithms a. b.

34. Most calculators (and software packages) have only two types of log keys, one for common logarithms (base 10) and one for natural logarithms (base e).

35. Example 1 Changing Bases Using Common Logarithms
Use a calculator.
Simplify.

36. HW:

37. HW:

38. 4.4 Logarithmic and Exponential Equations
A logarithmic equation involves the logarithm of an expression containing an unknown.
An exponential equation has the unknown appearing in an exponent.
There are two basic strategies for solving exponential or logarithmic equations.
The first is based on the One-to-One Properties and was used to solve simple exponential and logarithmic equations.
One-to-One Properties:
ax = ay if and only if x = y.
loga x = loga y if and only if x = y (a > 0 and a ≠ 1)
The second is based on the Inverse Properties.
Inverse Properties:
aloga x = x
loga ax = x

39. Example 1 – Solving Simple Equations

40. Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
An experiment was conducted with a particular type of small animal. The logarithm of the amount of oxygen consumed per hour was determined for a number of the animals and was plotted against the logarithms of the weights of the animals. It was found that
where y is the number of microliters of oxygen consumed per hour and x is the weight of the animal (in grams). Solve for y.

41. Solution: Chapter 4: Exponential and Logarithmic Functions
4.4 Logarithmic and Exponential Equations
Example 1 – Oxygen Composition
Solution:

47. HW
additional problems next slides

48. HW
additional problems next slides

49. Extra homework from different book: