Which Table Represents The Graph Of A Logarithmic Function With Both An $x$- And $y$-intercept? \[ \begin{tabular}{|c|c|} \hline X$ & Y Y Y \ \hline 3 & ∅ \emptyset ∅ \ \hline 4 & -15 \ \hline 5 & 0.585 \ \hline 6 & 1.322

Which Table Represents the Graph of a Logarithmic Function with Both an x- and y-Intercept?

In mathematics, a logarithmic function is a function that is the inverse of an exponential function. It is a fundamental concept in mathematics and has numerous applications in various fields, including science, engineering, and economics. A logarithmic function can be represented graphically, and understanding the characteristics of its graph is essential for analyzing and solving problems involving logarithmic functions.

Characteristics of Logarithmic Functions

A logarithmic function with base $b$ is defined as $y = \log_b x$. The graph of a logarithmic function has several key characteristics:

• Domain: The domain of a logarithmic function is all positive real numbers, i.e., $x > 0$.
• Range: The range of a logarithmic function is all real numbers, i.e., $y \in \mathbb{R}$.
• x-intercept: The x-intercept of a logarithmic function is the point where the graph intersects the x-axis, i.e., $y = 0$. Since the domain of a logarithmic function is all positive real numbers, the x-intercept is not defined.
• y-intercept: The y-intercept of a logarithmic function is the point where the graph intersects the y-axis, i.e., $x = 0$. The y-intercept of a logarithmic function is defined and is equal to $-\infty$.

Tables Representing Logarithmic Functions

We are given four tables representing different functions. We need to determine which table represents the graph of a logarithmic function with both an x- and y-intercept.

Analysis of Each Table

Let's analyze each table to determine which one represents the graph of a logarithmic function with both an x- and y-intercept.

Table 1

Table 1 does not represent the graph of a logarithmic function with both an x- and y-intercept. The y-intercept is not defined, and the x-intercept is not defined.

Table 2

Table 2 does not represent the graph of a logarithmic function with both an x- and y-intercept. The y-intercept is not defined, and the x-intercept is not defined.

Table 3

Table 3 represents the graph of a logarithmic function with base 10. The x-intercept is not defined, and the y-intercept is defined and is equal to $-\infty$.

Table 4

Table 4 represents the graph of a logarithmic function with base 10. The x-intercept is defined and is equal to 1, and the y-intercept is defined and is equal to $-\infty$.

Based on the analysis of each table, we can conclude that Table 4 represents the graph of a logarithmic function with both an x- and y-intercept.

• A logarithmic function with base $b$ is defined as $y = \log_b x$.
• The graph of a logarithmic function has several key characteristics, including a domain of all positive real numbers, a range of all real numbers, and an x-intercept that is not defined.
• The y-intercept of a logarithmic function is defined and is equal to $-\infty$.
• Table 4 represents the graph of a logarithmic function with both an x- and y-intercept.

Understanding the characteristics of logarithmic functions and their graphs is essential for analyzing and solving problems involving logarithmic functions. By recognizing the key characteristics of logarithmic functions, we can determine which table represents the graph of a logarithmic function with both an x- and y-intercept.
Logarithmic Functions: A Comprehensive Q&A Guide

Logarithmic functions are a fundamental concept in mathematics, and understanding their properties and characteristics is essential for analyzing and solving problems involving logarithmic functions. In this article, we will provide a comprehensive Q&A guide to logarithmic functions, covering their definition, characteristics, and applications.

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. It is defined as $y = \log_b x$, where $b$ is the base of the logarithm.

Q: What are the key characteristics of a logarithmic function?

A: The key characteristics of a logarithmic function are:

• Domain: The domain of a logarithmic function is all positive real numbers, i.e., $x > 0$.
• Range: The range of a logarithmic function is all real numbers, i.e., $y \in \mathbb{R}$.
• x-intercept: The x-intercept of a logarithmic function is the point where the graph intersects the x-axis, i.e., $y = 0$. Since the domain of a logarithmic function is all positive real numbers, the x-intercept is not defined.
• y-intercept: The y-intercept of a logarithmic function is the point where the graph intersects the y-axis, i.e., $x = 0$. The y-intercept of a logarithmic function is defined and is equal to $-\infty$.

Q: What is the difference between a logarithmic function and an exponential function?

A: A logarithmic function is the inverse of an exponential function. While an exponential function grows rapidly as the input increases, a logarithmic function grows slowly as the input increases.

Q: How do I determine the base of a logarithmic function?

A: To determine the base of a logarithmic function, you can use the following methods:

• Change of base formula: The change of base formula is $y = \log_b x = \frac{\log_a x}{\log_a b}$, where $a$ is any positive real number.
• Graphical method: You can use a graphical method to determine the base of a logarithmic function. Plot the graph of the function and identify the point where the graph intersects the y-axis.

Q: What are some common applications of logarithmic functions?

A: Logarithmic functions have numerous applications in various fields, including:

• Science: Logarithmic functions are used to model population growth, chemical reactions, and other scientific phenomena.
• Engineering: Logarithmic functions are used to design and analyze electronic circuits, communication systems, and other engineering applications.
• Economics: Logarithmic functions are used to model economic growth, inflation, and other economic phenomena.

Q: How do I solve logarithmic equations?

A: To solve logarithmic equations, you can use the following methods:

• Logarithmic properties: You can use logarithmic properties, such as the product rule and the quotient rule, to simplify the equation.
• Exponential form: You can convert the logarithmic equation to exponential form and solve for the variable.
• Graphical method: You can use a graphical method to solve the logarithmic equation.

Q: What are some common mistakes to avoid when working with logarithmic functions?

A: Some common mistakes to avoid when working with logarithmic functions include:

• Domain and range errors: Make sure to check the domain and range of the function before solving the equation.
• Exponential form errors: Make sure to convert the logarithmic equation to exponential form correctly.
• Graphical method errors: Make sure to plot the graph correctly and identify the point where the graph intersects the y-axis.

Logarithmic functions are a fundamental concept in mathematics, and understanding their properties and characteristics is essential for analyzing and solving problems involving logarithmic functions. By following the Q&A guide provided in this article, you can gain a deeper understanding of logarithmic functions and improve your problem-solving skills.