Select The Correct Answer.What Is The Solution Set To This Equation? Log ⁡ 4 ( X − 3 ) + Log ⁡ 4 ( X + 3 ) = 2 \log_4(x-3) + \log_4(x+3) = 2 Lo G 4 ​ ( X − 3 ) + Lo G 4 ​ ( X + 3 ) = 2 A. X = 16 X = 16 X = 16 B. X = 5 X = 5 X = 5 C. X = − 5 X = -5 X = − 5 And X = 5 X = 5 X = 5 D. X = − 16 X = -16 X = − 16 And X = 16 X = 16 X = 16

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving a specific logarithmic equation and provide a step-by-step guide on how to arrive at the solution. We will also discuss the importance of understanding logarithmic properties and how they can be applied to solve various types of equations.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithmic function is the inverse of the exponential function, and it's defined as:

$$\log_b(x) = y \iff b^y = x$$

where $b$ is the base of the logarithm, and $x$ is the input value.

One of the most important properties of logarithms is the product rule, which states that:

$$\log_b(xy) = \log_b(x) + \log_b(y)$$

This property can be used to simplify logarithmic expressions and solve equations.

Solving the Equation

The given equation is:

$$\log_4(x-3) + \log_4(x+3) = 2$$

To solve this equation, we can use the product rule of logarithms, which states that:

$$\log_b(xy) = \log_b(x) + \log_b(y)$$

We can rewrite the equation as:

$$\log_4((x-3)(x+3)) = 2$$

Using the product rule, we can simplify the left-hand side of the equation:

$$\log_4(x^2-9) = 2$$

Now, we can use the definition of logarithms to rewrite the equation in exponential form:

$$4^2 = x^2-9$$

Simplifying the equation, we get:

$$16 = x^2-9$$

Adding 9 to both sides of the equation, we get:

$$25 = x^2$$

Taking the square root of both sides of the equation, we get:

$$x = \pm 5$$

Therefore, the solution set to the equation is $x = -5$ and $x = 5$.

Conclusion

In this article, we solved a logarithmic equation using the product rule of logarithms and the definition of logarithms. We also discussed the importance of understanding logarithmic properties and how they can be applied to solve various types of equations. The solution set to the equation is $x = -5$ and $x = 5$.

Answer

The correct answer is:

• C. $x = -5$ and $x = 5$

Discussion

This equation can be solved using the properties of logarithms. The product rule of logarithms states that $\log_b(xy) = \log_b(x) + \log_b(y)$. This property can be used to simplify logarithmic expressions and solve equations.

The equation can be rewritten as $\log_4((x-3)(x+3)) = 2$. Using the product rule, we can simplify the left-hand side of the equation to $\log_4(x^2-9) = 2$.

Now, we can use the definition of logarithms to rewrite the equation in exponential form: $4^2 = x^2-9$. Simplifying the equation, we get $16 = x^2-9$. Adding 9 to both sides of the equation, we get $25 = x^2$.

Taking the square root of both sides of the equation, we get $x = \pm 5$. Therefore, the solution set to the equation is $x = -5$ and $x = 5$.

Key Takeaways

• The product rule of logarithms states that $\log_b(xy) = \log_b(x) + \log_b(y)$.
• The definition of logarithms states that $\log_b(x) = y \iff b^y = x$.
• The equation $\log_4(x-3) + \log_4(x+3) = 2$ can be solved using the product rule of logarithms and the definition of logarithms.
• The solution set to the equation is $x = -5$ and $x = 5$.

References

• [1] "Logarithmic Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html
• [2] "Logarithmic Properties" by Khan Academy. Retrieved from https://www.khanacademy.org/math/precalculus/precalc-logarithms/precalc-logarithmic-properties/v/logarithmic-properties

Additional Resources

• [1] "Logarithmic Equations" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/LogarithmicEquation.html
• [2] "Logarithmic Properties" by Purplemath. Retrieved from https://www.purplemath.com/modules/logrules.htm
  Logarithmic Equations: A Q&A Guide =====================================

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will provide a Q&A guide on logarithmic equations, covering various topics and concepts. Whether you're a student or a teacher, this guide will help you understand logarithmic equations and how to solve them.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The logarithmic function is the inverse of the exponential function, and it's defined as:

$$\log_b(x) = y \iff b^y = x$$

where $b$ is the base of the logarithm, and $x$ is the input value.

Q: What are the properties of logarithms?

A: The properties of logarithms are:

• Product rule: $\log_b(xy) = \log_b(x) + \log_b(y)$
• Quotient rule: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power rule: $\log_b(x^y) = y\log_b(x)$
• Definition of logarithms: $\log_b(x) = y \iff b^y = x$

Q: How do I solve a logarithmic equation?

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

1. Rewrite the equation in exponential form: Use the definition of logarithms to rewrite the equation in exponential form.
2. Simplify the equation: Simplify the equation by combining like terms and using the properties of logarithms.
3. Isolate the variable: Isolate the variable by moving all terms involving the variable to one side of the equation.
4. Solve for the variable: Solve for the variable by using algebraic techniques.

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

A: A logarithmic equation is an equation that involves a logarithmic function, while an exponential equation is an equation that involves an exponential function. The key difference between the two is that a logarithmic equation is the inverse of an exponential equation.

Q: Can you provide an example of a logarithmic equation?

A: Yes, here's an example of a logarithmic equation:

$$\log_4(x-3) + \log_4(x+3) = 2$$

To solve this equation, you can use the product rule of logarithms and the definition of logarithms.

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

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

• Not using the correct properties of logarithms: Make sure to use the correct properties of logarithms, such as the product rule and the quotient rule.
• Not rewriting the equation in exponential form: Make sure to rewrite the equation in exponential form using the definition of logarithms.
• Not simplifying the equation: Make sure to simplify the equation by combining like terms and using the properties of logarithms.
• Not isolating the variable: Make sure to isolate the variable by moving all terms involving the variable to one side of the equation.

Conclusion

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. By understanding the properties of logarithms and using the correct techniques, you can solve logarithmic equations with confidence. Remember to avoid common mistakes and to always check your work.

Additional Resources

• [1] "Logarithmic Equations" by Math Open Reference. Retrieved from https://www.mathopenref.com/logarithmic.html
• [2] "Logarithmic Properties" by Khan Academy. Retrieved from https://www.khanacademy.org/math/precalculus/precalc-logarithms/precalc-logarithmic-properties/v/logarithmic-properties
• [3] "Logarithmic Equations" by Wolfram MathWorld. Retrieved from https://mathworld.wolfram.com/LogarithmicEquation.html
• [4] "Logarithmic Properties" by Purplemath. Retrieved from https://www.purplemath.com/modules/logrules.htm