1. Chapter 3 Exponential and Logarithmic Functions
Pre-Calculus
Chapter 3
Exponential and Logarithmic Functions

2. 3.2 Logarithmic Functions
Objectives:
Recognize and evaluate logarithmic functions with base a.
Graph logarithmic functions.
Recognize, evaluate, and graph natural logarithmic functions.
Use logarithmic functions to model and solve real-life problems.

3. What is a logarithm? “Logarithm” comes from two Greek words
“Logos” = ratio
“Arithmos” = number
So, a logarithm is an exponent.

4. How Were Logarithms Developed?
Imagine a time before calculators….
In the 16th and 17th centuries, scientists made significant gains in the study of astronomy, navigation, and other areas. They wanted an easier (and less error-prone) way of performing calculations, esp. multiplication and division.
Two mathematicians, John Napier and Henry Briggs, are credited with developing logarithms to simplify calculations.

5. How Were Logarithms Used?
Recall the property of exponents: ax · ay = a(x + y)
Napier used this property to convert complicated, difficult multiplication problems into easier, less error-prone, addition problems.
For example, 4, x =
10m x n = 10(m + n)
where 3 < m < 4 and –1 < n < 0
Briggs spent a great deal of his life identifying values of y such that 10y = x. The value y is the logarithm. That is, y = log10 x. This data was organized into logarithmic tables.

6. A Sample Table Log. Exponent Form Number 100 1 log101 = 0 0.08720
100 1
log101 = 0
1.222
log = 0.087
2.459
log =
4.971
log =
101 10
log1010 = 1

7. Back to the Example Original statement: 4,971.26 x 0.2459
Rewrite in scientific notation:
x x x 10-1
Use values from table:
x x x 10-1
Simplify:
103 x
1, 000 x = 1,222
…Fortunately, we have calculators!

8. Definition of Logarithmic Function
The logarithmic function with base a is given by
f (x) = loga x
where x > 0, a > 0, and a ≠ 1.
That is,
y = loga x if and only if x = a y

9. Example 1 Rewrite each as an exponential expression and solve.
f (x) = log2 32
f (x) = log3 1
f (x) = log4 2
f (x) = log10 1/100

10. Example 2
Use your calculator to solve f (x) = log10 x at each value of x.
x = 10
x = 2.5
x = –2
x = ¼

11. Properties of Logarithms
loga 1 = 0 because a0 = 1.
loga a = 1 because a1 = a.
loga ax = x and a loga x = x Inverse Properties
If loga x = loga y, then x = y One-to-One Property

12. Example 3 Solve for x: log2 x = log2 3 Solve for x: log4 4 = x
Simplify: log5 5x
Simplify: 7 log7 14

13. Graph of Logarithmic Function
To sketch the graph of y = loga x, first graph y = ax, then graph its inverse.
Example: Graph f (x) = log2 x.
Construct a table of values for g(x) = 2x and plot the points.
The function f (x) = log2 x is the inverse of g(x) = 2x.
Note: Inverse functions are reflections in the line y = x.

14. Logarithmic Function The logarithmic function
is the inverse of the exponential function.
Many real-life phenomena with a slow rate of growth can be modeled by logarithmic functions.

15. Characteristics of the Logarithmic Function

16. Transformations of f (x) = loga x
The graph of g(x) = loga (x ± h) is a _________ shift of f.
The graph of h(x) = loga (x) ± k is a _________ shift of f.
The graph of j(x) = – loga (x) is a reflection of f ______ .
The graph of k(x) = loga (– x) is a reflection of f ______ .

17. Example 4
Each of the following functions is a transformation of the graph of f (x) = log10 x. Describe the transformation and graph.
g(x) = log10 (x – 1)
h (x) = 2 + log10 x

18. Natural Logarithmic Function
For x > 0,
y = ln x if and only if x = e y
The function given by
f (x) = loge x = ln x
is called the natural logarithmic function.

19. Example 5
Use your calculator to solve f (x) = ln x at each value of x.
x = 2
x = 0.3
x = –1

20. Properties of Natural Logs
ln 1 = 0 because e0 = 1.
ln e = 1 because e1 = e.
ln ex = x and e ln x = x Inverse Properties
If ln x = ln y, then x = y One-to-One Property

21. Example 6
Use properties of natural logs to rewrite each expresssion.

22. Graph of Natural Log Function
The graph of y = ln x is the reflection of the graph y = ex in the line y = x.

23. Domain of Log Functions
Find the domain of each function.
f (x) = ln (x – 2)
g(x) = ln (2 – x)
h(x) = ln x2

24. Transformations of f (x) = ln x
The graph of g(x) = ln (x ± h) is a ___________ shift of f.
The graph of h(x) = ln (x) ± k is a ___________ shift of f.
The graph of j(x) = – ln (x) is a reflection of f ________.
The graph of k(x) = ln (– x) is a reflection of f ________.

25. Example 7
Students participating in a psychology experiment attended several lectures on a subject and were given an exam. Every month for a year after the exam, the students were retested to see how much of the material they remembered. The average scores for the group are given by the human memory model
f (t) = 75 – 6 ln (t + 1), 0 ≤ t ≤ 12
where t is the time in months.
What was the average score on the original (t = 0) exam?
What was the average score at the end of t = 2 months?
What was the average score at the end of t = 6 months?

26. Homework 3.2
Worksheet 3.2