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

Mar 13, 2025 by ADMIN 205 views

**Understanding Logarithmic Functions and Their Graphs**

Introduction to Logarithmic Functions

Logarithmic functions are a fundamental concept in mathematics, and they play a crucial role in various fields such as science, engineering, and economics. A logarithmic function is a function that is the inverse of an exponential function. In other words, if we have an exponential function of the form $f(x) = a^x$ , then the corresponding logarithmic function is of the form $f(x) = \log_a(x)$ .

Characteristics of Logarithmic Functions

Logarithmic functions have several characteristics that distinguish them from other types of functions. One of the key characteristics of a logarithmic function is that it has a vertical asymptote at $x = 0$ . This means that the function approaches positive or negative infinity as $x$ approaches 0 from the right or left, respectively.

Another important characteristic of logarithmic functions is that they have a horizontal asymptote at $y = 0$ . This means that as $x$ approaches positive or negative infinity, the function approaches 0.

Graphs of Logarithmic Functions

The graph of a logarithmic function is a curve that approaches the vertical asymptote at $x = 0$ and the horizontal asymptote at $y = 0$ . The graph of a logarithmic function can have various shapes and forms, depending on the base of the logarithm and the range of the function.

Intercepts of Logarithmic Functions

A logarithmic function can have both $x$ -intercepts and $y$ -intercepts. An $x$ -intercept is a point on the graph where the function crosses the $x$ -axis, and a $y$ -intercept is a point on the graph where the function crosses the $y$ -axis.

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

To determine which table represents the graph of a logarithmic function with both an $x$ -intercept and a $y$ -intercept, we need to analyze the characteristics of the function and the shape of the graph.

Analyzing the Tables

Let's analyze the two tables given:

$x$	$y$
3	$\emptyset$
4	-15
5	$\emptyset$
$x$	$y$
---	---
1	0
2	1
3	2

Table 1 Analysis

From Table 1, we can see that the function has a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 0$ . The function also has an $x$ -intercept at $x = 4$ and a $y$ -intercept at $y = -15$ .

Table 2 Analysis

From Table 2, we can see that the function has a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = 0$ . The function also has an $x$ -intercept at $x = 2$ and a $y$ -intercept at $y = 1$ .

Conclusion

Based on the analysis of the tables, we can conclude that Table 1 represents the graph of a logarithmic function with both an $x$ -intercept and a $y$ -intercept.

Why Table 1 Represents the Graph of a Logarithmic Function with Both an $x$-intercept and $y$-intercept

Table 1 represents the graph of a logarithmic function with both an $x$ -intercept and a $y$ -intercept because it satisfies the characteristics of a logarithmic function. The function has a vertical asymptote at $x = 3$ and a horizontal asymptote at $y = 0$ , which are the defining characteristics of a logarithmic function. Additionally, the function has an $x$ -intercept at $x = 4$ and a $y$ -intercept at $y = -15$ , which are the points where the function crosses the $x$ -axis and the $y$ -axis, respectively.

Why Table 2 Does Not Represent the Graph of a Logarithmic Function with Both an $x$-intercept and $y$-intercept

Table 2 does not represent the graph of a logarithmic function with both an $x$ -intercept and a $y$ -intercept because it does not satisfy the characteristics of a logarithmic function. The function does not have a vertical asymptote at $x = 1$ and a horizontal asymptote at $y = 0$ , which are the defining characteristics of a logarithmic function. Additionally, the function does not have an $x$ -intercept at $x = 2$ and a $y$ -intercept at $y = 1$ , which are the points where the function crosses the $x$ -axis and the $y$ -axis, respectively.

Conclusion

Q: What is a logarithmic function?

A: A logarithmic function is a function that is the inverse of an exponential function. In other words, if we have an exponential function of the form $f(x) = a^x$ , then the corresponding logarithmic function is of the form $f(x) = \log_a(x)$ .

Q: What are the characteristics of a logarithmic function?

A: The characteristics of a logarithmic function include:

A vertical asymptote at $x = 0$
A horizontal asymptote at $y = 0$
An $x$ -intercept and a $y$ -intercept

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

A: The main difference between a logarithmic function and an exponential function is that a logarithmic function is the inverse of an exponential function. In other words, if we have an exponential function of the form $f(x) = a^x$ , then the corresponding logarithmic function is of the form $f(x) = \log_a(x)$ .

Q: How do I determine if a function is logarithmic or exponential?

A: To determine if a function is logarithmic or exponential, you can use the following criteria:

If the function has a vertical asymptote at $x = 0$ and a horizontal asymptote at $y = 0$ , it is likely a logarithmic function.
If the function has a horizontal asymptote at $y = 0$ and a vertical asymptote at $x = 0$ , it is likely an exponential function.

Q: What is the domain and range of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers, and the range is all real numbers.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use the following steps:

Plot the vertical asymptote at $x = 0$
Plot the horizontal asymptote at $y = 0$
Plot the $x$ -intercept and the $y$ -intercept
Use a graphing calculator or software to plot the function

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

$\log_2(x)$
$\log_3(x)$
$\log_4(x)$
$\log_5(x)$

Q: How do I evaluate a logarithmic function?

A: To evaluate a logarithmic function, you can use the following steps:

Use the definition of a logarithmic function: $\log_a(x) = y \iff a^y = x$
Use a calculator or software to evaluate the function

Q: What are some real-world applications of logarithmic functions?

A: Some real-world applications of logarithmic functions include:

Finance: logarithmic functions are used to calculate interest rates and investment returns
Science: logarithmic functions are used to model population growth and decay
Engineering: logarithmic functions are used to design and optimize systems

Q: How do I solve logarithmic equations?

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

Use the definition of a logarithmic function: $\log_a(x) = y \iff a^y = x$
Use algebraic manipulations to isolate the variable
Use a calculator or software to solve the 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:

Confusing the base of the logarithm with the exponent
Failing to check the domain and range of the function
Using the wrong formula or method to evaluate the function

Conclusion

In conclusion, logarithmic functions are an important concept in mathematics, and they have many real-world applications. By understanding the characteristics, graphing, and evaluating logarithmic functions, you can solve a wide range of problems and make informed decisions in various fields.