# 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, =

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

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 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.

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.

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

• 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.

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