MTH120 College Algebra Chapter 6.4

 6.4 Graphs of Logarithmic Functions

Graphs of logarithmic functions display specific characteristics that make them distinctive. The general form of a logarithmic function is $�={\log }_{�}(�)$, where $�$ is the base of the logarithm. Below are some key features of the graphs of logarithmic functions:

1. Domain and Range:

   • The domain of a logarithmic function is $(0,\mathrm{\infty })$, meaning it's defined for positive values of $�$.
   • The range of a logarithmic function is all real numbers, which means the graph extends both upward and downward.

2. Horizontal Asymptote:

   • Logarithmic functions have a horizontal asymptote at $�=0$. The graph approaches but never reaches the x-axis.

3. Intercepts:

   • The graph of $�={\log }_{�}(�)$ has a y-intercept at $(1,0)$ because ${\log }_{�}(1)=0$.
   • Logarithmic functions do not have x-intercepts since they are only defined for positive values of $�$.

4. Increasing and Unbounded Growth:

   • Logarithmic functions are always increasing. As $�$ gets larger, $�$ increases without bound.
   • This is in contrast to exponential functions, where the graph increases rapidly.

5. Graph Shape:

   • The shape of the graph is concave upward, meaning it opens in an upward direction.
   • The steepness of the curve depends on the base ($�$) of the logarithm. A larger base results in a steeper curve.

6. Asymptotic Behavior:

   • As $�$ approaches 0 from the right, the graph becomes steeper, and ${\lim }_{�\to 0^{+}}{\log }_{�}(�)=-\mathrm{\infty }$.
   • As $�$ increases, the graph approaches but never crosses the line $�=0$.

7. Logarithmic Transformations:

   • Like other functions, logarithmic functions can be transformed. For example, shifting the graph horizontally or vertically or stretching/compressing it.

8. Base Effects:

   • The base ($�$) of the logarithm affects the graph's steepness. Larger bases result in steeper graphs.

Below is an example of the graph of $�=\log (�)$ (base 10):

This graph demonstrates many of the characteristics mentioned above. As $�$ approaches 0 from the right, $�$ becomes increasingly negative, indicating that the logarithm grows without bound as $�$ approaches 0. The graph is concave upward, and it never crosses the line $�=0$, representing the horizontal asymptote.

The domain of a logarithmic function, $�={\log }_{�}(�)$, represents all the values of $�$ for which the function is defined. In the context of logarithmic functions, the domain has a specific constraint: $�$ must be greater than 0, as logarithms are undefined for non-positive arguments.

Here are some examples of finding the domain of logarithmic functions with explanations:

1. Example 1: $�={\log }_2(�)$

   • In this case, the base of the logarithm is 2, and $�$ must be greater than 0.
   • So, the domain is $(0,\mathrm{\infty })$, which means all positive real numbers.

2. Example 2: $�={\log }_{10}(�)$

   • Here, the base is 10, which is commonly referred to as the common logarithm.
   • As with all logarithmic functions, the argument $�$ must be greater than 0.
   • So, the domain is $(0,\mathrm{\infty })$, meaning all positive real numbers.

3. Example 3: $�=\ln (�)$

   • This is the natural logarithm with base $�$.
   • As always, the argument $�$ must be greater than 0 for logarithms to be defined.
   • Therefore, the domain is $(0,\mathrm{\infty })$.

4. Example 4: $�={\log }_3(�-2)$

   • In this example, the argument inside the logarithm is $�-2$.
   • For this function to be defined, $�-2$ must be greater than 0.
   • This means $�-2>0$.
   • Solve for $�$: $�>2$.
   • So, the domain is $(2,\mathrm{\infty })$.

5. Example 5: $�=\log (�+1)$

   • Similarly, for this function to be defined, $�+1$ must be greater than 0.
   • This implies $�+1>0$.
   • Solve for $�$: (x > -1.
   • Therefore, the domain is $(-1,\mathrm{\infty })$.

In summary, when finding the domain of a logarithmic function, remember that $�$ must be greater than 0. This is a fundamental constraint, and the domain is usually expressed in interval notation to indicate all positive real numbers. If there are transformations or shifts applied to the argument, you need to consider those in determining the domain.

Graphing logarithmic functions is a valuable skill that helps you understand the behavior of these functions. Logarithmic functions have specific characteristics, such as asymptotes, increasing nature, and transformations. Here are some examples of graphing logarithmic functions with explanations:

Example 1: Graph $�={\log }_2(�)$

• This is a logarithmic function with a base of 2.
• Start by making a table of values. Choose various positive values for $�$ and calculate the corresponding values of $�$.
• For instance:
  • If $�=1$, $�={\log }_2(1)=0$.
  • If $�=2$, $�={\log }_2(2)=1$.
  • If $�=4$, $�={\log }_2(4)=2$.
• Continue this process to create more points. You'll notice that the function is increasing and never touches the x-axis.
• There's a vertical asymptote at $�=0$ because ${\log }_2(�)$ is undefined for non-positive values.
• The graph gets steeper as $�$ increases, reflecting the exponential growth of the logarithm.

Example 2: Graph $�={\log }_{10}(�)$

• This is the common logarithm with a base of 10.
• It behaves similarly to the previous example but is scaled differently.
• It has the same vertical asymptote at $�=0$.
• The graph is also increasing, but it grows more slowly compared to the base-2 logarithm.

Example 3: Graph $�=\ln (�)$

• This is the natural logarithm with a base of $�$.
• The graph exhibits the same behavior as the previous examples but is characterized by a steeper growth rate.
• It also has a vertical asymptote at $�=0$.
• The graph never touches the x-axis and is always increasing.

Example 4: Graph $�={\log }_3(�-1)$

• Here, the function involves a shift to the right by 1 unit.
• To graph it, start by shifting the graph of $�={\log }_3(�)$ one unit to the right.
• The vertical asymptote that was at $�=0$ in the original graph is now at $�=1$.
• The graph is still increasing and never touches the x-axis.

Example 5: Graph $�={\log }_2(2�)$

• This function represents a horizontal compression by a factor of 2.
• The vertical asymptote remains at $�=0$, but the graph becomes steeper as $�$ increases.
• To create the graph, you can use the properties of logarithms to understand how the compression affects the function.

Remember that the behavior of logarithmic functions, including the presence of a vertical asymptote, the increasing nature of the graph, and the impact of transformations, can vary based on the base and any shifts or compressions applied to the function. Graphing these functions helps you visualize these characteristics and understand their behavior.

The parent function of a logarithmic function is often denoted as $�(�)={\log }_{�}(�)$, where $�$ is the base of the logarithm. The graph of this function has several characteristic features:

1. Domain and Range:

   • The domain of the logarithmic function is all positive real numbers, meaning $�$ is greater than 0, or $(0,\mathrm{\infty })$.
   • The range is all real numbers, which means $�$ can be any real number.

2. Vertical Asymptote:

   • The graph has a vertical asymptote at $�=0$. This means the graph gets arbitrarily close to, but never touches, the y-axis as $�$ approaches 0 from the right.

3. Increasing Nature:

   • The graph is always increasing. As $�$ gets larger, $�$ increases without bound.
   • This is in contrast to exponential functions, which increase rapidly.

4. Horizontal Behavior:

   • There is no horizontal asymptote for logarithmic functions. The graph extends indefinitely in the positive and negative y-directions.

5. Intercepts:

   • The graph has a y-intercept at $(1,0)$ because ${\log }_{�}(1)=0$.
   • Logarithmic functions do not have x-intercepts since they are only defined for positive values of $�$.

6. Shape:

   • The shape of the graph is concave upward, meaning it opens in an upward direction. It is symmetric with respect to the line $�=1$.
   • The steepness of the curve depends on the base ($�$) of the logarithm. A larger base results in a steeper curve.

7. Base Effects:

   • The base ($�$) of the logarithm affects the graph's steepness. Larger bases result in steeper graphs.
   • For example, $�(�)={\log }_{10}(�)$ has a gentler slope compared to $�(�)={\log }_2(�)$, as the base 10 is smaller than the base 2.

8. Transformations:

   • Logarithmic functions can be transformed by shifting them horizontally or vertically, as well as stretching or compressing them.
   • For example, $�(�)={\log }_{�}(�-�)$ represents a horizontal shift by $�$ units to the right.

These characteristics help you understand the behavior and appearance of the graph of the parent logarithmic function $�(�)={\log }_{�}(�)$. When working with specific logarithmic functions, you can apply these features along with transformations to create more complex graphs.

Graphing transformations of logarithmic functions involves applying shifts, stretches, and compressions to the parent function $�(�)={\log }_{�}(�)$ in order to create new functions. Here are some examples of how these transformations affect the graph:

Parent Function: $�(�)={\log }_{�}(�)$

1. Horizontal Shift (Right/Left):

   • A horizontal shift by $�$ units to the right is represented by $�(�)={\log }_{�}(�-�)$.
   • A horizontal shift by $�$ units to the left is represented by $�(�)={\log }_{�}(�+�)$.
   • The effect of a horizontal shift is to move the graph horizontally while maintaining the other characteristics of the parent function.

2. Vertical Shift (Up/Down):

   • A vertical shift upward by $�$ units is represented by $�(�)={\log }_{�}(�)+�$.
   • A vertical shift downward by $�$ units is represented by $�(�)={\log }_{�}(�)-�$.
   • The effect of a vertical shift is to move the entire graph up or down without changing its shape.

3. Stretch or Compression (Vertical):

   • A vertical stretch by a factor of $�$ is represented by $�(�)=�\cdot {\log }_{�}(�)$.
   • A vertical compression by a factor of $�$ is represented by $�(�)=\frac{1}{�}\cdot {\log }_{�}(�)$.
   • A vertical stretch makes the graph steeper, while a vertical compression makes it gentler.

4. Reflect about the x-axis:

   • Reflecting the graph about the x-axis is represented by $�(�)=-{\log }_{�}(�)$.
   • This transformation flips the graph upside down.

5. Reflect about the y-axis:

   • Reflecting the graph about the y-axis is represented by $�(�)={\log }_{�}(-�)$.
   • This transformation reflects the graph to the left of the y-axis.

Here are examples of transformed logarithmic functions:

Example 1: $�(�)={\log }_2(�-2)$

• This is a horizontal shift to the right by 2 units. The vertical asymptote moves to $�=2$.

Example 2: $�(�)={\log }_5(�)+3$

• This is a vertical shift upward by 3 units. The entire graph is lifted by 3 units.

Example 3: $�(�)=2{\log }_3(�)$

• This is a vertical stretch by a factor of 2. The graph becomes steeper.

Example 4: $�(�)=-{\log }_{10}(�)$

• This reflects the graph about the x-axis, flipping it upside down.

Example 5: $�(�)={\log }_2(-�)$

• This reflects the graph about the y-axis, making it appear to the left of the y-axis.

Understanding these transformations allows you to graph more complex logarithmic functions by applying these changes to the parent function $�(�)={\log }_{�}(�)$.

A horizontal shift of a logarithmic function $�(�)={\log }_{�}(�)$ is achieved by modifying the argument of the logarithm, which shifts the entire graph left or right along the x-axis. Here's how you can graph a horizontally shifted logarithmic function:

General Form: $�(�)={\log }_{�}(�-�)$

• A horizontal shift to the right by $�$ units is represented by subtracting $�$ from the $�$ value inside the logarithm.

Steps to Graph $�(�)={\log }_{�}(�-�)$:

1. Identify the Parent Function: Start with the parent logarithmic function $�(�)={\log }_{�}(�)$.

2. Determine the Horizontal Shift: If $�$ is positive, it's a shift to the right. If $�$ is negative, it's a shift to the left.

3. Find the New Vertical Asymptote: The vertical asymptote, which is normally at $�=0$ for the parent function, will now be at $�=�$ for the shifted function.

4. Plot Key Points:

   • Plot the vertical asymptote at $�=�$.
   • Choose additional points on the graph by picking various values of $�$ greater than $�$.
   • Calculate the corresponding $�$ values using the logarithmic function. For each point, $�={\log }_{�}(�-�)$.
   • Plot these points.

5. Analyze the Graph:

   • The graph should exhibit the same general characteristics as the parent logarithmic function (increasing, vertical asymptote, etc.).
   • It is shifted horizontally by $�$ units to the right if $�$ is positive.

6. Label the Graph:

   • Label the vertical asymptote at $�=�$ and indicate the direction of the shift.

Example: Graph $�(�)={\log }_2(�-2)$

• This represents a horizontal shift of the parent function $�(�)={\log }_2(�)$ to the right by 2 units.
• The vertical asymptote, which is normally at $�=0$, is now at $�=2$.

Here's how to graph it:

1. Identify the parent function: $�(�)={\log }_2(�)$.
2. Determine the horizontal shift: $�=2$ (to the right).
3. Find the new vertical asymptote: $�=2$.
4. Plot key points:
   • Vertical asymptote at $�=2$.
   • Choose $�=3$, $�=4$, and $�=5$ to calculate corresponding $�$ values.
   • For example, at $�=3$, $�={\log }_2(3-2)={\log }_2(1)=0$.
5. Analyze the graph: The graph is a horizontally shifted logarithmic function, opening to the right. It approaches, but never touches, the vertical asymptote at $�=2$.
6. Label the graph: Mark the vertical asymptote at $�=2$.

This will give you the graph of the logarithmic function $�(�)={\log }_2(�-2)$ that has been horizontally shifted 2 units to the right compared to the parent function.

When you have a logarithmic function with the form $�(�)={\log }_{�}(�+�)$, you're dealing with a horizontal translation (shift) of the parent logarithmic function $�(�)={\log }_{�}(�)$. The value of $�$ determines how far left or right the graph is shifted. Here's how to graph this translation:

General Form: $�(�)={\log }_{�}(�+�)$

• If $�$ is positive, it's a shift to the left.
• If $�$ is negative, it's a shift to the right.

Steps to Graph $�(�)={\log }_{�}(�+�)$:

1. Identify the Parent Function: Begin with the parent logarithmic function $�(�)={\log }_{�}(�)$.

2. Determine the Horizontal Shift: Depending on the sign of $�$, determine whether it's a left or right shift.

3. Find the New Vertical Asymptote: The vertical asymptote, normally at $�=0$ for the parent function, is now shifted to $�=-�$.

4. Plot Key Points:

   • Plot the vertical asymptote at $�=-�$.
   • Select additional points for the graph by choosing various values of $�$ greater than $-�$.
   • Calculate the corresponding $�$ values using the logarithmic function: $�={\log }_{�}(�+�)$.
   • Plot these points.

5. Analyze the Graph:

   • The graph should maintain the same general characteristics of the parent logarithmic function (increasing, vertical asymptote, etc.).
   • It is shifted horizontally by $�$ units in the specified direction.

6. Label the Graph:

   • Label the vertical asymptote at $�=-�$.

Example: Graph $�(�)={\log }_2(�+3)$

• This represents a horizontal shift of the parent function $�(�)={\log }_2(�)$ to the left by 3 units.
• The vertical asymptote, which is normally at $�=0$, is now at $�=-3$.

Here's how to graph it:

1. Identify the parent function: $�(�)={\log }_2(�)$.
2. Determine the horizontal shift: $�=3$ (to the left).
3. Find the new vertical asymptote: $�=-3$.
4. Plot key points:
   • Vertical asymptote at $�=-3$.
   • Choose $�=0$, $�=1$, and $�=2$ to calculate corresponding $�$ values.
   • For example, at $�=1$, $�={\log }_2(1+3)={\log }_2(4)=2$.
5. Analyze the graph: The graph is a horizontally shifted logarithmic function to the left by 3 units. It approaches but never touches the vertical asymptote at $�=-3$.
6. Label the graph: Mark the vertical asymptote at $�=-3$.

This will give you the graph of the logarithmic function $�(�)={\log }_2(�+3)$ that has been horizontally shifted 3 units to the left compared to the parent function.

Graphing a vertical shift of a logarithmic function $�={\log }_{�}(�)$ involves modifying the function by adding or subtracting a constant term to the entire function, resulting in a vertical shift. Here's how to graph a vertical shift:

General Form: $�={\log }_{�}(�)+�$

• If $�$ is positive, it's a shift upward.
• If $�$ is negative, it's a shift downward.

Steps to Graph $�={\log }_{�}(�)+�$:

1. Identify the Parent Function: Begin with the parent logarithmic function $�={\log }_{�}(�)$.

2. Determine the Vertical Shift: Depending on the sign of $�$, decide whether it's an upward or downward shift.

3. Plot Key Points:

   • Select key points for the graph of the parent function.
   • Calculate the corresponding $�$ values for the shifted function by adding or subtracting $�$.

4. Analyze the Graph:

   • The graph should maintain the same general characteristics of the parent logarithmic function (increasing, vertical asymptote, etc.).
   • It is shifted vertically by $�$ units in the specified direction.

5. Label the Graph:

   • If $�$ is positive, label the shift as "upward" and indicate the shift amount.
   • If $�$ is negative, label the shift as "downward" and indicate the shift amount.

Example: Graph $�={\log }_2(�)+3$

• This represents a vertical shift of the parent function $�={\log }_2(�)$ upward by 3 units.

Here's how to graph it:

1. Identify the parent function: $�={\log }_2(�)$.
2. Determine the vertical shift: $�=3$ (upward).
3. Plot key points:
   • Select key points from the graph of the parent function.
   • Calculate the corresponding $�$ values for the shifted function by adding 3 to the $�$ values.
4. Analyze the graph: The graph is a logarithmic function shifted upward by 3 units.
5. Label the graph: Indicate the upward shift by 3 units.

This will give you the graph of the logarithmic function $�={\log }_2(�)+3$ that has been vertically shifted upward by 3 units compared to the parent function.

Graphing stretches and compressions of a logarithmic function $�={\log }_{�}(�)$ involves modifying the function by multiplying or dividing the logarithmic term by a constant, resulting in a vertical stretch or compression. Here's how to graph stretches and compressions:

General Forms:

• Vertical Stretch: $�=�\cdot {\log }_{�}(�)$ (where $�>1$)
• Vertical Compression: $�=\frac{1}{�}\cdot {\log }_{�}(�)$ (where $0<�<1$)

The value of $�$ determines the degree of stretching or compression.

Steps to Graph Stretches and Compressions:

1. Identify the Parent Function: Begin with the parent logarithmic function $�={\log }_{�}(�)$.

2. Determine the Stretch or Compression: The value of $�$ will determine whether it's a vertical stretch or compression:

   • If $�>1$, it's a vertical stretch.
   • If $0<�<1$, it's a vertical compression.

3. Plot Key Points:

   • Select key points for the graph of the parent function.
   • Calculate the corresponding $�$ values for the stretched or compressed function by applying the vertical transformation.
   • When stretching, multiply $�$ values by $�$.
   • When compressing, divide $�$ values by $�$.

4. Analyze the Graph:

   • The graph should maintain the same general characteristics of the parent logarithmic function (increasing, vertical asymptote, etc.).
   • It is stretched or compressed vertically by a factor of $�$.

5. Label the Graph:

   • Indicate whether it's a stretch or compression and provide the value of $�$.

Example 1: Graph $�=2\cdot {\log }_2(�)$

• This represents a vertical stretch of the parent function $�={\log }_2(�)$ by a factor of 2.

Here's how to graph it:

1. Identify the parent function: $�={\log }_2(�)$.
2. Determine the stretch: $�=2$.
3. Plot key points:
   • Select key points from the graph of the parent function.
   • Calculate the corresponding $�$ values for the stretched function by multiplying the $�$ values by 2.
4. Analyze the graph: The graph is a logarithmic function stretched vertically by a factor of 2.
5. Label the graph: Indicate the vertical stretch by a factor of 2.

Example 2: Graph $�=\frac{1}{3}\cdot {\log }_2(�)$

• This represents a vertical compression of the parent function $�={\log }_2(�)$ by a factor of 3.

Here's how to graph it:

1. Identify the parent function: $�={\log }_2(�)$.
2. Determine the compression: $�=\frac{1}{3}$.
3. Plot key points:
   • Select key points from the graph of the parent function.
   • Calculate the corresponding $�$ values for the compressed function by dividing the $�$ values by 3.
4. Analyze the graph: The graph is a logarithmic function compressed vertically by a factor of 3.
5. Label the graph: Indicate the vertical compression by a factor of 3.

These steps will help you graph logarithmic functions that have been stretched or compressed compared to the parent function $�={\log }_{�}(�)$.

When you have a logarithmic function with the form $�(�)=�\cdot {\log }_{�}(�)$, where $�>0$, you are dealing with a vertical stretch (if $�>1$) or vertical compression (if $0<�<1$) of the parent logarithmic function $�(�)={\log }_{�}(�)$. This transformation modifies the amplitude of the function. Here's how to graph it:

General Form: $�(�)=�\cdot {\log }_{�}(�)$

• If $�>1$, it's a vertical stretch.
• If $0<�<1$, it's a vertical compression.

Steps to Graph $�(�)=�\cdot {\log }_{�}(�)$:

1. Identify the Parent Function: Begin with the parent logarithmic function $�(�)={\log }_{�}(�)$.

2. Determine the Vertical Stretch or Compression: The value of $�$ determines whether it's a stretch or compression:

   • If $�>1$, it's a vertical stretch.
   • If $0<�<1$, it's a vertical compression.

3. Plot Key Points:

   • Select key points for the graph of the parent function.
   • Calculate the corresponding $�$ values for the stretched or compressed function by applying the vertical transformation.
   • When stretching, multiply $�$ values by $�$.
   • When compressing, divide $�$ values by $�$.

4. Analyze the Graph:

   • The graph should maintain the same general characteristics of the parent logarithmic function (increasing, vertical asymptote, etc.).
   • It is stretched or compressed vertically by a factor of $�$.

5. Label the Graph:

   • Indicate whether it's a stretch or compression and provide the value of $�$.

Example 1: Graph $�(�)=2\cdot {\log }_2(�)$

• This represents a vertical stretch of the parent function $�(�)={\log }_2(�)$ by a factor of 2.

Here's how to graph it:

1. Identify the parent function: $�(�)={\log }_2(�)$.
2. Determine the vertical stretch: $�=2$.
3. Plot key points:
   • Select key points from the graph of the parent function.
   • Calculate the corresponding $�$ values for the stretched function by multiplying the $�$ values by 2.
4. Analyze the graph: The graph is a logarithmic function stretched vertically by a factor of 2.
5. Label the graph: Indicate the vertical stretch by a factor of 2.

Example 2: Graph $�(�)=\frac{1}{3}\cdot {\log }_2(�)$

• This represents a vertical compression of the parent function $�(�)={\log }_2(�)$ by a factor of $\frac{1}{3}$.

Here's how to graph it:

1. Identify the parent function: $�(�)={\log }_2(�)$.
2. Determine the vertical compression: $�=\frac{1}{3}$.
3. Plot key points:
   • Select key points from the graph of the parent function.
   • Calculate the corresponding $�$ values for the compressed function by dividing the $�$ values by $\frac{1}{3}$.
4. Analyze the graph: The graph is a logarithmic function compressed vertically by a factor of $\frac{1}{3}$.
5. Label the graph: Indicate the vertical compression by a factor of $\frac{1}{3}$.

These steps will help you graph logarithmic functions that have been stretched or compressed compared to the parent function $�(�)={\log }_{�}(�)$.

Graphing reflections of a logarithmic function $�(�)={\log }_{�}(�)$ involves reflecting the graph across either the x-axis or the y-axis. Here's how to graph these reflections:

General Forms:

1. Reflection about the X-Axis: $�(�)=-{\log }_{�}(�)$

   • This reflects the graph across the x-axis, flipping it upside down.

2. Reflection about the Y-Axis: $�(�)={\log }_{�}(-�)$

   • This reflects the graph across the y-axis, making it appear on the left side of the y-axis.

Steps to Graph Reflections:

Reflection about the X-Axis: $�(�)=-{\log }_{�}(�)$

1. Identify the Parent Function: Start with the parent logarithmic function $�(�)={\log }_{�}(�)$.

2. Apply the Reflection: When you have $�(�)=-{\log }_{�}(�)$, it's a reflection about the x-axis.

3. Plot Key Points:

   • Select key points for the graph of the parent function.
   • Calculate the corresponding $�$ values for the reflected function by changing the sign of the $�$ values (i.e., if $�$ is positive, make it negative, and vice versa).

4. Analyze the Graph:

   • The graph will be a reflection of the parent logarithmic function about the x-axis, appearing upside down.

5. Label the Graph:

   • Indicate the reflection about the x-axis.

Reflection about the Y-Axis: $�(�)={\log }_{�}(-�)$

1. Identify the Parent Function: Start with the parent logarithmic function $�(�)={\log }_{�}(�)$.

2. Apply the Reflection: When you have $�(�)={\log }_{�}(-�)$, it's a reflection about the y-axis.

3. Plot Key Points:

   • Select key points for the graph of the parent function.
   • Calculate the corresponding $�$ values for the reflected function by changing the sign of the $�$ values (i.e., if $�$ is positive, make it negative, and vice versa).

4. Analyze the Graph:

   • The graph will be a reflection of the parent logarithmic function about the y-axis, appearing on the left side of the y-axis.

5. Label the Graph:

   • Indicate the reflection about the y-axis.

Example 1: Reflection about the X-Axis Graph $�(�)=-{\log }_2(�)$

• This represents a reflection of the parent function $�(�)={\log }_2(�)$ about the x-axis.

Example 2: Reflection about the Y-Axis Graph $�(�)={\log }_2(-�)$

• This represents a reflection of the parent function $�(�)={\log }_2(�)$ about the y-axis.

To graph these reflections, follow the steps for each type of reflection, and remember to label the graph to indicate the type of reflection you've applied.

Translations of the logarithmic function involve shifting or modifying the graph of the parent logarithmic function $�(�)={\log }_{�}(�)$. These translations can be summarized as follows:

1. Horizontal Shift (Left/Right):

   • A horizontal shift to the right by $�$ units is represented by $�(�)={\log }_{�}(�-�)$.
   • A horizontal shift to the left by $�$ units is represented by $�(�)={\log }_{�}(�+�$).

2. Vertical Shift (Up/Down):

   • A vertical shift upward by $�$ units is represented by $�(�)={\log }_{�}(�)+�$.
   • A vertical shift downward by $�$ units is represented by $�(�)={\log }_{�}(�)-�$.

3. Stretch or Compression (Vertical):

   • A vertical stretch by a factor of $�$ is represented by $�(�)=�\cdot {\log }_{�}(�)$.
   • A vertical compression by a factor of $�$ is represented by $�(�)=\frac{1}{�}\cdot {\log }_{�}(�)$.

4. Reflections:

   • A reflection about the x-axis is represented by $�(�)=-{\log }_{�}(�)$.
   • A reflection about the y-axis is represented by $�(�)={\log }_{�}(-�$).

Understanding these translations allows you to manipulate the parent logarithmic function $�(�)={\log }_{�}(�)$ to create various functions with different shifts, stretches, compressions, and reflections. These transformations help you analyze and graph logarithmic functions in a variety of scenarios.

For the following exercises, sketch the graph of the indicated function. 1. f(x)=log6(x+1) 2. f(x)=3log(x) 3. f(x)=ln(−x) 4. g(x)=log(3x+13)+5 5. g(x)=log(4−6x)+7 6. h(x)=−11ln(x+3)−5

Let's sketch the graphs of the indicated functions:

1. $�(�)={\log }_6(�+1)$

   • This is a logarithmic function with a horizontal shift to the left by 1 unit.
   • The vertical asymptote is at $�=-1$.
   • The graph is increasing and approaches the asymptote.

2. $�(�)=3\log (�)$

   • This is a logarithmic function with a vertical stretch by a factor of 3.
   • The parent function is $�(�)=\log (�)$, which has a vertical asymptote at $�=0$.
   • The graph is steeper than the parent function and still has an asymptote at $�=0$.

3. $�(�)=\ln (-�)$

   • This is a natural logarithmic function with a reflection about the y-axis and a horizontal shift to the right by 0.
   • The domain is restricted to $�<0$ because the natural logarithm is only defined for positive values.
   • The graph is decreasing and approaches negative infinity as $�$ approaches 0 from the left.

4. $�(�)=\log (3�+13)+5$

   • This is a logarithmic function with a horizontal shift to the left by 4.333 units and a vertical shift upward by 5 units.
   • The vertical asymptote is at $�=-4.333$.
   • The graph is increasing and approaches the asymptote.

5. $�(�)=\log (4-6�)+7$

   • This is a logarithmic function with a horizontal shift to the right by 0.6667 units and a vertical shift upward by 7 units.
   • The vertical asymptote is at $�=0.6667$.
   • The graph is increasing and approaches the asymptote.

6. $ℎ(�)=-11\ln (�+3)-5$

   • This is a natural logarithmic function with a horizontal shift to the left by 3 units, a vertical stretch by a factor of 11, and a vertical shift downward by 5 units.
   • The domain is restricted to $�>-3$ to keep the natural logarithm defined.
   • The graph is decreasing and approaches negative infinity as $�$ approaches -3 from the right.

These are rough sketches of the functions based on the described transformations. You may want to use graphing software or calculators for more precise graphs.