Problem 76 - College Algebra and Calculus: An Applied Approach

Question

Use a graphing utility to graph the function. Be sure to use an appropriate viewing window.\(f(x)=\log (x-1)\)

Step-by-Step Solution

Answer

The graph of the function \(f(x)=\log (x-1)\) starts at a vertical asymptote on the line \(x=1\) and slowly increases as \(x\) values increase. The graph is under the x-axis for \(x

Step 1: Identify the Function Type

The function \(f(x)=\log (x-1)\) is a logarithmic function, translated one unit to the right in the x-axis from the standard parent function \(f(x)=\log(x)\). Logarithmic functions have a domain of \(x>0\), and for this function, the domain is \(x>1\) because of the translation. Therefore, the graph should cover values of \(x\) that are greater than 1.

Step 2: Graphing the Function

Use a graphing utility to plot the function \(f(x)=\log (x-1)\). Select an appropriate window to clearly depict the main characteristics of the function. Start the x-axis from 0 or a small negative value (to show the vertical asymptote clearly) and extend it till 3 or 4 to get enough coverage. Set your \(y\) values lower limit as \(-5\) and upper limit as \(5\) for a clear view of the function behavior.

Step 3: Verify the Graph

The resulting graph should show a curve starting from a vertical asymptote on the line \(x=1\) and increasing slowly as \(x\) values increase. It will stay under the x-axis for \(x

Key Concepts

Logarithmic Function Properties

Understanding the properties of logarithmic functions is crucial in graphing them correctly. A logarithmic function is the inverse of an exponential function and has the form of \( f(x) = \log_b(x) \), where \( b \) is the base. The function \( f(x) = \log (x-1) \) that you're working with has particular characteristics:

• The function is undefined when \( x \) is less than or equal to 0, hence it's domain-exclusive to positive real numbers.
• It has a vertical asymptote at \( x = 1 \) due to the transformation; this is the x-value the graph approaches but never touches.
• The range is all real numbers, meaning the output can be any number on the y-axis.
• As \( x \) increases in value, the slope of the curve decreases, evidenced by the flattening out of the graph.
• The graph will cross the y-axis at the point where \( x \) is equal to the base of the logarithm, which would be \( b+1 \) in our transformed function, provided the base is \( e \), the natural logarithm base.

These properties are foundational in understanding how changes in the function will affect the graph.

Choosing a Graphical Viewer Window

Choosing the correct viewing window on your graphical utility is essential to accurately display the key features of a logarithmic function. For the function \( f(x)=\log (x-1) \), we need to ensure that the vertical asymptote at \( x=1 \) is visible. Here is how you can select an appropriate window:

• Set the x-min to be a little less than the asymptote - perhaps at 0 or a negative number if your graphing utility permits. This makes the asymptote visible at \( x=1 \).
• Choose an x-max that is sufficiently large to observe the nature of the function as it flattens - possibly between 3 and 4.
• For the y-axis, a common range is from \( -5 \) to \( 5 \) to display a comprehensive view of where the function increases and decreases.

By carefully selecting these window parameters, you ensure that all relevant details of the logarithmic function are presented within the viewing screen.

Transformation of Logarithmic Functions

The transformation of logarithmic functions involves shifting, stretching, or reflecting the graph by altering its equation. With our example \( f(x)=\log (x-1) \), there is a horizontal shift to the right by 1 unit from the parent function \( \log(x) \).

Here's how transformations impact the function graph:

• A horizontal shift occurs when you add or subtract a constant from the input variable \( x \). The subtraction of 1 in our function causes the graph to shift right by 1 unit.
• Vertical shifts happen when a constant is added or subtracted to the output of the function \( f(x) \).
• Reflections occur when the input or output is multiplied by negative one, effectively 'flipping' the graph over the x-axis or y-axis.
• Vertical stretching or compressing occurs when the output is multiplied by a factor greater or less than one, respectively.

Understanding these transformations allows you to predict how the graph of a function will look before even plotting it.

Domain of Logarithmic Functions

The domain of a logarithmic function, such as \( f(x)=\log (x-1) \), is vitally important when graphing. The domain consists of all the values that \( x \) can take. For a basic logarithm function \( \log(x) \), the domain is \( x > 0 \), because the log of a negative number or zero is undefined.

In our specific example, since the function is translated one unit to the right, the domain shifts accordingly to \( x > 1 \). This means graphically, you won't plot anything to the left of \( x=1 \), because the function doesn't exist there—it's not within the set of allowable \( x \) values. By recognizing this, you avoid common errors where portions of the graph are mistakenly plotted where the function does not exist.

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Other exercises in this chapter

Problem 76

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(2-6 \ln x=10\)

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Problem 76

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\ln \frac{x

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Problem 76

Hospital Employment The numbers of people \(E\) (in thousands) employed in hospitals from 1999 to 2005 can be modeled by \(E=3331(1.0182)^{t}, \quad 9 \leq t \l

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Problem 77

Solve the logarithmic equation algebraically. Approximate the result to three decimal places.\(\ln x-\ln (x+1)=2\)

View solution

Key Concepts

Logarithmic Function Properties

Understanding the properties of logarithmic functions is crucial in graphing them correctly. A logarithmic function is the inverse of an exponential function and has the form of \( f(x) = \log_b(x) \), where \( b \) is the base. The function \( f(x) = \log (x-1) \) that you're working with has particular characteristics:

• The function is undefined when \( x \) is less than or equal to 0, hence it's domain-exclusive to positive real numbers.
• It has a vertical asymptote at \( x = 1 \) due to the transformation; this is the x-value the graph approaches but never touches.
• The range is all real numbers, meaning the output can be any number on the y-axis.
• As \( x \) increases in value, the slope of the curve decreases, evidenced by the flattening out of the graph.
• The graph will cross the y-axis at the point where \( x \) is equal to the base of the logarithm, which would be \( b+1 \) in our transformed function, provided the base is \( e \), the natural logarithm base.

These properties are foundational in understanding how changes in the function will affect the graph.

Choosing a Graphical Viewer Window

Choosing the correct viewing window on your graphical utility is essential to accurately display the key features of a logarithmic function. For the function \( f(x)=\log (x-1) \), we need to ensure that the vertical asymptote at \( x=1 \) is visible. Here is how you can select an appropriate window:

• Set the x-min to be a little less than the asymptote - perhaps at 0 or a negative number if your graphing utility permits. This makes the asymptote visible at \( x=1 \).
• Choose an x-max that is sufficiently large to observe the nature of the function as it flattens - possibly between 3 and 4.
• For the y-axis, a common range is from \( -5 \) to \( 5 \) to display a comprehensive view of where the function increases and decreases.

By carefully selecting these window parameters, you ensure that all relevant details of the logarithmic function are presented within the viewing screen.

Transformation of Logarithmic Functions

The transformation of logarithmic functions involves shifting, stretching, or reflecting the graph by altering its equation. With our example \( f(x)=\log (x-1) \), there is a horizontal shift to the right by 1 unit from the parent function \( \log(x) \).

Here's how transformations impact the function graph:

• A horizontal shift occurs when you add or subtract a constant from the input variable \( x \). The subtraction of 1 in our function causes the graph to shift right by 1 unit.
• Vertical shifts happen when a constant is added or subtracted to the output of the function \( f(x) \).
• Reflections occur when the input or output is multiplied by negative one, effectively 'flipping' the graph over the x-axis or y-axis.
• Vertical stretching or compressing occurs when the output is multiplied by a factor greater or less than one, respectively.

Understanding these transformations allows you to predict how the graph of a function will look before even plotting it.

Domain of Logarithmic Functions

The domain of a logarithmic function, such as \( f(x)=\log (x-1) \), is vitally important when graphing. The domain consists of all the values that \( x \) can take. For a basic logarithm function \( \log(x) \), the domain is \( x > 0 \), because the log of a negative number or zero is undefined.

In our specific example, since the function is translated one unit to the right, the domain shifts accordingly to \( x > 1 \). This means graphically, you won't plot anything to the left of \( x=1 \), because the function doesn't exist there—it's not within the set of allowable \( x \) values. By recognizing this, you avoid common errors where portions of the graph are mistakenly plotted where the function does not exist.