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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 x-value where it is undefined and near which the function’s values become very large positive or negative numbers.
  • x-intercept – A point where the graph of a function crosses the x-axis.

Review

Remember…
  • Form of a Logarithmic Function: F(x)=log_b x
  • When you find the natural logarithm of a number, you are using base e, which is approximately equal to 2.718.
    • F(x)=ln\;x is the same as F(x)=log_e x
  • 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: F(x)=log_b x, does not exist for b = 0 or b = 1
    • Why can’t the base be zero in a logarithmic function?
      • log_0 x 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
  1. Set your calculator to scientific mode.
  2. Enter the number of which you want to take the natural logarithm.
  3. Press the “ln” button.
  •  Example: find the natural log: F(x)=ln\; 3
    • 3, ln, =
    • Answer: 1.10 (rounded)

 

Graphs of Logarithmic Functions
  • F(x)=log_b x
    • Increasing: b > 1
    • Decreasing: 0 < b < 1
    • y-axis: vertical asymptote
    • x-intercept: (1, 0)
    • Domain: all positive real numbers
    • Range: all real numbers
Alg2B 9.6 Log Graphs Ex

 

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

Alg2B 9.6 Log V Shift

 

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

Alg2B 9.6 Log H Shift

 

Vertical Stretching or Compressing
  • Reflections
    • Multiplying a function by a negative number flips the graph across the x-axis.
    • A reflection across the x-axis 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.

Alg2B 9.6 Stretch Compress

 

Horizontal Stretching or Compressing
  • Reflections
    • Multiplying the input variable by a negative number, flips the function across the y-axis and changes the domain.
    • Otherwise, the range and domain stay the same for a vertical stretch or compression.
  • Stretch or Compress
    • To horizontally stretch a logarithmic function, you multiply the input variable by a number between zero and 1 (a positive fraction).
    • To horizontally compress it, you multiply the input variable by a number greater than 1.
  • Domain & Range
    • The domain of this reflected graph is x < 0
    • The range does not change.
  • Examples
    • G(x)=log(\frac{1}{4}x) is an example of a horizontal stretch.
    • H(x)=log(7x) is an example of a horizontal compression.

Alg2B 9.6 H Stretch Compress

 

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

Alg2B 9.6 Translations


  • Step-by-Step Translation Example
    • F(x)=-3(log_10 x)-4
      • It is a reflection of the parent function over the x-axis.
      • The parent function was translated down.
      • It is a vertical stretch of the parent function.

Alg2B 9.6 Translations Ex

 

Comparison Chart
Alg2B 9.6 Comparison Chart

Examples

  • Ex 1. What is the domain of the logarithmic function given below?
  • F(x)=log_3 x
    • Answer: x > 0
  • Ex 2. What is the range of the logarithmic function given below?
  • F(x)=log_0.3 x
    • Answer: All real numbers
  • Ex 3. Which facts are true for the graph of the function below?
  • F(x)=log_3 x
    • Answers:
      • The x-intercept is (1, 0).
      • It is increasing.
  • Ex 4. It is true that the domain of F(x)=log_b x is the set of all positive real numbers.
  • Ex 5. It is true that the range of F(x)=log_b x is the set of all real numbers.
  • Ex 6. Which facts are true for the graph of the function below?
  • F(x)=log_0.427 x
    • Answers:
      • It is decreasing
      • The range is all real numbers
      • The domain is x > 0
      • The x-intercept is (1,0)
  • Ex 7. It is true that the function F(x)=log0.9 x is decreasing.
  • Ex 8. For what values of b will F(x)=log_b x be an increasing function?
    • b > 1
  • Ex 9. For what values of b will F(x)=log_b x be a decreasing function?
    • 0 < b < 1

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