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Graph the function. g(x) = log, (r)

Graph the function. g(x) = log, (r)

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Graphing Logarithmic Functions** Function: \( g(x) = \log_6(x) \) Below are two graphical representations of the function \( g(x) = \log_6(x) \). **Graph 1:** In the first graph, the function \( g(x) = \log_6(x) \) is plotted. The x-axis (horizontal axis) and the y-axis (vertical axis) are labeled clearly. The graph shows the following characteristics: - The graph is a curve that increases slowly and passes through the point (1, 0), since the logarithm of 1 is always 0 regardless of the base. - The curve approaches the y-axis but never actually touches it (asymptotic behavior). - As \( x \) increases, the values of \( g(x) \) continue to increase, though at a decreasing rate. **Graph 2:** This graph is identical to the first one, and it also represents \( g(x) = \log_6(x) \) with the same characteristics as described above. This redundancy might be due to an error in including two identical plots. In both graphs, an arrow indicates the direction of the function, emphasizing the behavior as \( x \) increases. Key insights into the graphs: - The logarithmic function grows slowly and never produces negative values for positive \( x \). - The function is not defined for non-positive \( x \) values. - The graph exhibits a vertical asymptote at \( x = 0 \). These characteristics are typical for logarithmic functions with different bases, except for modifications in growth rate depending on the base value. Here, the base is 6.
Transcribed Image Text:**Graphing Logarithmic Functions** Function: \( g(x) = \log_6(x) \) Below are two graphical representations of the function \( g(x) = \log_6(x) \). **Graph 1:** In the first graph, the function \( g(x) = \log_6(x) \) is plotted. The x-axis (horizontal axis) and the y-axis (vertical axis) are labeled clearly. The graph shows the following characteristics: - The graph is a curve that increases slowly and passes through the point (1, 0), since the logarithm of 1 is always 0 regardless of the base. - The curve approaches the y-axis but never actually touches it (asymptotic behavior). - As \( x \) increases, the values of \( g(x) \) continue to increase, though at a decreasing rate. **Graph 2:** This graph is identical to the first one, and it also represents \( g(x) = \log_6(x) \) with the same characteristics as described above. This redundancy might be due to an error in including two identical plots. In both graphs, an arrow indicates the direction of the function, emphasizing the behavior as \( x \) increases. Key insights into the graphs: - The logarithmic function grows slowly and never produces negative values for positive \( x \). - The function is not defined for non-positive \( x \) values. - The graph exhibits a vertical asymptote at \( x = 0 \). These characteristics are typical for logarithmic functions with different bases, except for modifications in growth rate depending on the base value. Here, the base is 6.
### Logarithmic Functions: Graphs and Behavior The following diagrams illustrate the graph of the logarithmic function \( g(x) = \log_6{x} \). Each of the three graphs depicts the same function. #### Graph Explanation - **Function Notation:** - The function represented is \( g(x) = \log_6{x} \). This denotes a logarithmic function with base 6. - **Axes and Origin:** - Each graph is plotted on a Cartesian coordinate system with an \( x \)-axis (horizontal) and a \( y \)-axis (vertical). - The origin (0,0) is the point where the \( x \)-axis and \( y \)-axis intersect. - **Graph Characteristics:** - The graph starts from negative infinity as \( x \) approaches zero from the right (but never actually reaches zero). - The graph passes through the point (1,0), which is a characteristic of logarithmic functions since \(\log_b{1} = 0\) for any base \( b \) (here \( b = 6 \)). - As \( x \) increases, the value of \( g(x) \) increases but at a decreasing rate, reflecting the property of logarithmic growth. #### Graphs Included 1. **First Graph:** - The graph of \( g(x) = \log_6{x} \) is illustrated with a magenta curve. It displays the function's characteristic shape, starting steeply below and to the left of the \( y \)-axis and flattening out to the right. 2. **Second Graph:** - This graph also shows \( g(x) = \log_6{x} \) with similar properties, illustrating the logarithmic curve's behavior from negative infinity on the \( y \)-axis towards infinity on the \( x \)-axis. 3. **Third Graph:** - The third depiction is another instance of \( g(x) = \log_6{x} \), reinforcing the consistent properties of logarithmic growth regardless of the base. The repeated depiction emphasizes understanding the logarithmic function's fundamental nature, focusing on its behavior near the vertical asymptote at \( x = 0 \) and its unbounded growth along the \( x \)-axis.
Transcribed Image Text:### Logarithmic Functions: Graphs and Behavior The following diagrams illustrate the graph of the logarithmic function \( g(x) = \log_6{x} \). Each of the three graphs depicts the same function. #### Graph Explanation - **Function Notation:** - The function represented is \( g(x) = \log_6{x} \). This denotes a logarithmic function with base 6. - **Axes and Origin:** - Each graph is plotted on a Cartesian coordinate system with an \( x \)-axis (horizontal) and a \( y \)-axis (vertical). - The origin (0,0) is the point where the \( x \)-axis and \( y \)-axis intersect. - **Graph Characteristics:** - The graph starts from negative infinity as \( x \) approaches zero from the right (but never actually reaches zero). - The graph passes through the point (1,0), which is a characteristic of logarithmic functions since \(\log_b{1} = 0\) for any base \( b \) (here \( b = 6 \)). - As \( x \) increases, the value of \( g(x) \) increases but at a decreasing rate, reflecting the property of logarithmic growth. #### Graphs Included 1. **First Graph:** - The graph of \( g(x) = \log_6{x} \) is illustrated with a magenta curve. It displays the function's characteristic shape, starting steeply below and to the left of the \( y \)-axis and flattening out to the right. 2. **Second Graph:** - This graph also shows \( g(x) = \log_6{x} \) with similar properties, illustrating the logarithmic curve's behavior from negative infinity on the \( y \)-axis towards infinity on the \( x \)-axis. 3. **Third Graph:** - The third depiction is another instance of \( g(x) = \log_6{x} \), reinforcing the consistent properties of logarithmic growth regardless of the base. The repeated depiction emphasizes understanding the logarithmic function's fundamental nature, focusing on its behavior near the vertical asymptote at \( x = 0 \) and its unbounded growth along the \( x \)-axis.
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