The Concept of Logarithmic Functions

Oct 13, 2025 by recoveryshake.com 18 views

Understanding the Problem

The given function is $5 \log(x - 3)$ . To find the x-intercept, we need to determine the value of x when the function equals zero. In other words, we need to solve the equation $5 \log(x - 3) = 0$ .

The Concept of Logarithmic Functions

A logarithmic function is the inverse of an exponential function. It is defined as the power to which a base number must be raised to produce a given value. In this case, the base number is 10, and the function is $5 \log(x - 3)$ .

Solving the Equation

To solve the equation $5 \log(x - 3) = 0$ , we need to isolate the logarithmic term. We can do this by dividing both sides of the equation by 5, which gives us $\log(x - 3) = 0$ .

Using the Definition of Logarithms

By definition, a logarithmic function is equal to zero when its argument is equal to 1. Therefore, we can rewrite the equation $\log(x - 3) = 0$ as $x - 3 = 1$ .

Solving for x

Now we can solve for x by adding 3 to both sides of the equation, which gives us $x = 4$ .

Conclusion

The x-intercept of the function $5 \log(x - 3)$ is the point where the function equals zero. We found that this point is $(4, 0)$ .

Answer

The correct answer is D. $(4, 0)$ .

Why the Other Options are Incorrect

Let's take a look at the other options:

A. $(1, 0)$ : This point is not on the graph of the function, because when x = 1, the argument of the logarithmic function is $1 - 3 = -2$ , which is negative and therefore not in the domain of the function.
B. $(5, 0)$ : This point is not on the graph of the function, because when x = 5, the argument of the logarithmic function is $5 - 3 = 2$ , which is not equal to 1.
C. $(-3, 0)$ : This point is not on the graph of the function, because when x = -3, the argument of the logarithmic function is $-3 - 3 = -6$ , which is negative and therefore not in the domain of the function.

Final Thoughts

Frequently Asked Questions

Q: What is the x-intercept of a logarithmic function?

A: The x-intercept of a logarithmic function is the point where the function equals zero. In other words, it is the value of x that makes the function equal to zero.

Q: How do I find the x-intercept of a logarithmic function?

A: To find the x-intercept of a logarithmic function, you need to solve the equation $f(x) = 0$ , where $f(x)$ is the function. This involves isolating the logarithmic term and using the definition of logarithms to solve for x.

Q: What is the definition of a logarithmic function?

A: A logarithmic function is the inverse of an exponential function. It is defined as the power to which a base number must be raised to produce a given value.

Q: How do I know if a point is on the graph of a logarithmic function?

A: To determine if a point is on the graph of a logarithmic function, you need to check if the argument of the logarithmic function is positive and non-zero. If the argument is negative or zero, the point is not on the graph.

Q: What is the domain of a logarithmic function?

A: The domain of a logarithmic function is all real numbers greater than zero. In other words, the argument of the logarithmic function must be positive and non-zero.

Q: How do I solve an equation involving a logarithmic function?

A: To solve an equation involving a logarithmic function, you need to isolate the logarithmic term and use the definition of logarithms to solve for x. You can also use properties of logarithms, such as the product rule and the quotient rule, to simplify the equation.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

Not checking if the argument of the logarithmic function is positive and non-zero
Not using the definition of logarithms to solve for x
Not simplifying the equation using properties of logarithms
Not checking if the solution is in the domain of the function

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you need to plot points on the graph using a table of values. You can also use a graphing calculator or software to graph the function.

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

A: Logarithmic functions have many real-world applications, including:

Modeling population growth and decay
Analyzing financial data, such as stock prices and interest rates
Studying the behavior of chemical reactions
Modeling the spread of diseases

Q: How do I choose the correct base for a logarithmic function?

A: The choice of base for a logarithmic function depends on the problem you are trying to solve. Common bases include 10, e, and 2.

Q: What is the relationship between logarithmic and exponential functions?

A: Logarithmic and exponential functions are inverse functions. This means that if $f(x) = a^x$ , then $f^{-1}(x) = \log_a(x)$ .

Q: How do I use logarithmic functions to solve problems involving growth and decay?

A: Logarithmic functions can be used to model growth and decay by using the properties of logarithms to solve for the rate of growth or decay.

Q: What are some common logarithmic functions?

A: Some common logarithmic functions include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving finance?

A: Logarithmic functions can be used to model financial data, such as stock prices and interest rates, by using the properties of logarithms to solve for the rate of return or the interest rate.

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

A: Logarithmic functions have many real-world applications in science, including:

Modeling the behavior of chemical reactions
Analyzing the spread of diseases
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving engineering?

A: Logarithmic functions can be used to model engineering problems, such as the behavior of electrical circuits and the design of mechanical systems.

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

A: Some common mistakes to avoid when graphing logarithmic functions include:

Not checking if the argument of the logarithmic function is positive and non-zero
Not using a table of values to plot points on the graph
Not checking if the solution is in the domain of the function

Q: How do I use logarithmic functions to solve problems involving technology?

A: Logarithmic functions can be used to model technological problems, such as the behavior of computer algorithms and the design of communication systems.

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

A: Logarithmic functions have many real-world applications in economics, including:

Modeling the behavior of economic systems
Analyzing the spread of economic data
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving medicine?

A: Logarithmic functions can be used to model medical problems, such as the behavior of diseases and the design of medical devices.

Q: What are some common logarithmic functions used in medicine?

A: Some common logarithmic functions used in medicine include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving environmental science?

A: Logarithmic functions can be used to model environmental problems, such as the behavior of climate change and the design of sustainable systems.

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

A: Logarithmic functions have many real-world applications in environmental science, including:

Modeling the behavior of climate change
Analyzing the spread of environmental data
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving social science?

A: Logarithmic functions can be used to model social science problems, such as the behavior of population growth and decay and the design of social systems.

Q: What are some common logarithmic functions used in social science?

A: Some common logarithmic functions used in social science include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving business?

A: Logarithmic functions can be used to model business problems, such as the behavior of stock prices and the design of financial systems.

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

A: Logarithmic functions have many real-world applications in business, including:

Modeling the behavior of stock prices
Analyzing the spread of financial data
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving government?

A: Logarithmic functions can be used to model government problems, such as the behavior of population growth and decay and the design of public policy.

Q: What are some common logarithmic functions used in government?

A: Some common logarithmic functions used in government include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving education?

A: Logarithmic functions can be used to model educational problems, such as the behavior of student learning and the design of educational systems.

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

A: Logarithmic functions have many real-world applications in education, including:

Modeling the behavior of student learning
Analyzing the spread of educational data
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving law?

A: Logarithmic functions can be used to model legal problems, such as the behavior of population growth and decay and the design of legal systems.

Q: What are some common logarithmic functions used in law?

A: Some common logarithmic functions used in law include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving architecture?

A: Logarithmic functions can be used to model architectural problems, such as the behavior of building design and the design of urban systems.

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

A: Logarithmic functions have many real-world applications in architecture, including:

Modeling the behavior of building design
Analyzing the spread of architectural data
Studying the behavior of population growth and decay

Q: How do I use logarithmic functions to solve problems involving engineering?

A: Logarithmic functions can be used to model engineering problems, such as the behavior of electrical circuits and the design of mechanical systems.

Q: What are some common logarithmic functions used in engineering?

A: Some common logarithmic functions used in engineering include:

$\log_{10}(x)$
$\log_e(x)$
$\log_2(x)$

Q: How do I use logarithmic functions to solve problems involving computer science?

A: Log