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9.6 – Graphs of Logarithmic Functions
Key Terms
 Vertical Asymptote – A vertical line that the graph of a function approaches but never intersects.
 A function has a vertical asymptote at every xvalue where it is undefined and near which the function’s values become very large positive or negative numbers.
 xintercept – A point where the graph of a function crosses the xaxis.
Review
Remember… 
 Form of a Logarithmic Function:
 When you find the natural logarithm of a number, you are using base e, which is approximately equal to 2.718.
 is the same as
 All exponential functions (and their inverses, called logarithms) are curves, not lines.
 In an exponential function (and the inverse, called a logarithm), the base “b” is positive but not equal to 1.
 Therefore, a logarithmic function, which takes the form: , does not exist for b = 0 or b = 1
 Why can’t the base be zero in a logarithmic function?
 answers the question “How many times do you need to multiply zero to get x?”
 No matter how many times you multiply zero, you always get zero; so, b cannot be 0.

Notes
Calculate the Natural Log 
 Set your calculator to scientific mode.
 Enter the number of which you want to take the natural logarithm.
 Press the “ln” button.

 Example: find the natural log:
 3, ln, =
 Answer: 1.10 (rounded)

Graphs of Logarithmic Functions 

 Increasing: b > 1
 Decreasing: 0 < b < 1
 yaxis: vertical asymptote
 xintercept: (1, 0)
 Domain: all positive real numbers
 Range: all real numbers


Vertical Shifts 
 Vertical shifts do not affect the domain or range.

Horizontal Translations 
 A horizontal shift changes the domain of a logarithmic function.

Vertical Stretching or Compressing 
 Reflections
 Multiplying a function by a negative number flips the graph across the xaxis.
 A reflection across the xaxis does not affect the domain or range.
 Stretching & Compressing
 To vertically stretch a logarithmic function, you multiply the function by a number greater than 1.
 To vertically compress it, you multiply by a number between zero and 1 (a positive fraction).
 Domain & Range
 The range and domain stay the same for a vertical stretch or compression.

Translations (All Together) 
 You know that changes to the input mean horizontal translations, dilations, or reflections.
 StepbyStep Translation Example

 It is a reflection of the parent function over the xaxis.
 The parent function was translated down.
 It is a vertical stretch of the parent function.

Comparison Chart 

Examples
 Ex 1. What is the domain of the logarithmic function given below?


 Ex 2. What is the range of the logarithmic function given below?


 Ex 3. Which facts are true for the graph of the function below?

 Answers:
 The xintercept is (1, 0).
 It is increasing.

 Ex 4. It is true that the domain of is the set of all positive real numbers.

 Ex 5. It is true that the range of is the set of all real numbers.

 Ex 6. Which facts are true for the graph of the function below?

 Answers:
 It is decreasing
 The range is all real numbers
 The domain is x > 0
 The xintercept is (1,0)

 Ex 7. It is true that the function is decreasing.

 Ex 8. For what values of b will be an increasing function?

 Ex 9. For what values of b will be a decreasing function?

Important!
Practice (Apex Study 9.6)
 Try practice problems on Pgs 2, 14, 17, 18
 Mandatory: write and answer problems on Pg 19
 1 Quiz
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