1. Logarithmic Functions & Their Graphs
Section 3.2

2. Log Functions & Their Graphs
In the previous section, we worked with exponential functions.
What did the graph of these functions look like?

3. Log Functions & Their Graphs
Earlier in the year, we covered “inverse functions”
Do exponential functions have an inverse?
By looking at the graphs of exponential functions, we notice that every graph passes the horizontal line test.
Therefore, all exponential functions have an inverse

4. Log Functions & Their Graphs
The inverse of an exponential function with base a is called the logarithmic function with base a
For x > 0, a > 0 and a ≠ 1

5. Log Functions & Their Graphs
In other words:
really means that a raised to the power of y is equal to x
The log button on your calculator refers to the Log base 10
This is referred to as the Common Logarithm

6. Log Functions & Their Graphs
Another common logarithm is the Log base e
This is referred to as the Natural Logarithmic Function
This function is denoted:

7. Log Functions & Their Graphs
Write the following logarithms in exponential form.

8. Log Functions & Their Graphs
Write the exponential equations in log form

9. Log Functions & Their Graphs
Evaluate the following logarithms:
Since a raised to the
power of zero is equal to 1,
Since a raised to the
power of one is equal to a
= 0
= 1

10. Log Functions & Their Graphs
Now that we know the definition of a logarithmic function, we can start to evaluate basic logarithms.
What is this question asking?
2 raised to what power equals 8?
2³= 8
x = 3

11. Log Functions & Their Graphs
Evaluate the following logarithms:

12. Log Functions & Their Graphs
Properties of Logarithms

13. Log Functions & Their Graphs
Using these properties, we can simplify different logarithmic functions.
= x
From our third property, we can evaluate this log function to be equal to x.

14. Log Functions & Their Graphs
Use the properties of logarithms to evaluate or simplify the following expressions.

15. Log Functions & Their Graphs
In conclusion, what does the following statement mean?
“10 raised to the power of y is equal to z”

16. Logarithmic Functions & Their Graphs
Section 3.2

17. Log Functions & Their Graphs
Yesterday, we went over the basic definition of logarithms.
Remember, they are truly defined as the inverse of an exponential function.

18. Graphs of Log Functions
Fill in the following table and sketch the graph of the function f(x) for:
f(x) = x -2 -1 1 2 3

19. Graphs of Log Functions
Remember that the function is actually the inverse of the exponential function
To graph inverses, switch the x and y values
This is a reflection across the line y = x

20. Graphs of Log Functions
Fill in the following table and sketch the graph of the function f(x) for: -2 -1 1 2 3

21. Graphs of Log Functions
The nature of this curve is typical of the curves of logarithmic functions.
They have one x-intercept and one vertical asymptote
Reflection of the exponential curve across the line y = x

22. Graphs of Log Functions
Basic characteristics of the log curves
Domain: (0, ∞)
Range: (- ∞, ∞)
x-intercept at (1, 0)
Increasing
1-1 → the function has an inverse
y-axis is a vertical asymptote
Continuous

23. Graphs of Log Functions
Much like we had shifts in exponential curves, the log curves have shifts and reflections as well
Graphing will shift the curve 1 unit to the right
Graphing will shift the curve vertically up 2 units

24. Graphs of Log Functions
Much like we had shifts in exponential curves, the log curves have shifts and reflections as well
Graphing will reflect the curve over the vertical asymptote
Graphing will reflect the curve over the x-axis

25. Graphs of Log Functions
Sketch a graph of the following functions.

26. Graphs of Log Functions
Domain:
(3, ∞)
x-intercept:
(4, 0)
Asymptote:
x = 3

27. Graphs of Log Functions
Domain:
(1, ∞)
x-intercept:
( , 0)
Asymptote:
x = 1

28. Graphs of Log Functions
Domain:
(- ∞, 3)
x-intercept:
(2, 0)
Asymptote:
x = 3

29. Graphs of Log Functions
Notice that the first piece of information we have been gathering on the graphs is the domain.
For x > 0, a > 0 and a ≠ 1
This means that whatever value is in the place of x must be positive

30. Graphs of Log Functions
What would the domain of this function be?
(0, ∞)
→ - x > 0
→ x < 0
→ The domain would be: (- ∞, 0)

31. Graphs of Log Functions
Find the domain of the following logarithms. a) b) c)

32. Applications
The model below approximates the length of a home mortgage of $150,000 at 8% interest in terms of the monthly payment. In the model, t is the number of years of the mortgage and x is the monthly payment in dollars.

33. Applications
Use this model to approximate the length of a mortgage if the monthly payment is $1,300. By putting $1,300 in for x, you should get a time of 18.4 years

34. Applications
How much would you end up paying in interest using this same example?
Paying $1,300 a month for 18.4 years
→ Pay a total of (18.4) (1,300) =
Therefore, interest would be equal to $137,040.
$287,040

35. Applications
Using this same model, approximate the length of a mortgage when the monthly payment is: a) $1, and b) $1,254.68

36. Applications
a) $1, and b) $1, What would the difference in amount paid be for each of these mortgages?
30 years
20 years

37. Applications
A principal P, invested at 6% interest compounded continuously, increases to an amount K times the original principal after t years, where t is given by: How long will it take the original investment to double? By putting in 2 for K, we get t = years