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 = 5 X=5 X = 5 B. X = − 16 X=-16 X = − 16 And X = 16 X=16 X = 16 C. X = 16 X=16 X = 16 D. X = − 5 X=-5 X = − 5 And X = 5 X=5 X = 5

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.

The Equation

The given equation is:

$\log _4(x-3)+\log _4(x+3)=2$

This equation involves logarithms with base 4 and two variables, x-3 and x+3, inside the logarithms. Our goal is to find the solution set for this equation, which means we need to determine the values of x that satisfy the equation.

Step 1: Apply the Product Property of Logarithms

To simplify the equation, we can apply the product property of logarithms, which states that:

$\log _a(b) + \log _a(c) = \log _a(bc)$

Using this property, we can rewrite the equation as:

$\log _4((x-3)(x+3))=2$

Step 2: Simplify the Equation

Now, we can simplify the equation by evaluating the expression inside the logarithm:

$(x-3)(x+3) = x^2 - 9$

So, the equation becomes:

$\log _4(x^2 - 9)=2$

Step 3: Exponentiate Both Sides

To get rid of the logarithm, we can exponentiate both sides of the equation with base 4:

$4^{\log _4(x^2 - 9)} = 4^2$

Using the property of logarithms that states $a^{\log _a(x)} = x$, we can simplify the left-hand side of the equation:

$x^2 - 9 = 16$

Step 4: Solve for x

Now, we can solve for x by adding 9 to both sides of the equation:

$x^2 = 25$

Taking the square root of both sides, we get:

$x = \pm 5$

Conclusion

In conclusion, the solution set to the equation $\log _4(x-3)+\log _4(x+3)=2$ is $x = -5$ and $x = 5$. This means that both values of x satisfy the equation.

Importance of Logarithmic Properties

Understanding logarithmic properties is crucial in solving logarithmic equations. The product property of logarithms, which we applied in Step 1, is a fundamental concept that helps simplify logarithmic expressions. By applying this property, we can rewrite the equation in a more manageable form, making it easier to solve.

Real-World Applications

Logarithmic equations have numerous real-world applications in various fields, including physics, engineering, and economics. For example, logarithmic equations can be used to model population growth, chemical reactions, and financial transactions. By understanding how to solve logarithmic equations, we can better analyze and interpret data in these fields.

Common Mistakes to Avoid

When solving logarithmic equations, it's essential to avoid common mistakes, such as:

• Not applying the product property of logarithms
• Not simplifying the equation properly
• Not checking for extraneous solutions

By being aware of these potential pitfalls, we can ensure that our solutions are accurate and reliable.

Final Thoughts

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

A: The main difference between a logarithmic equation and an exponential equation is the direction of the operation. In a logarithmic equation, we have a logarithm (log) of a number, while in an exponential equation, we have a number raised to a power (e.g., 2^x). Logarithmic equations involve finding the power to which a base is raised to obtain a given number, while exponential equations involve finding the result of raising a base to a given power.

Q: How do I know which base to use when solving a logarithmic equation?

A: The base of a logarithmic equation is usually given in the problem statement. If the base is not specified, we can assume it is 10 (common logarithm) or e (natural logarithm). In this case, we can use the change-of-base formula to convert the logarithm to a common or natural logarithm.

Q: What is the change-of-base formula?

A: The change-of-base formula is:

$\log _a(x) = \frac{\log _b(x)}{\log _b(a)}$

This formula allows us to convert a logarithm with base a to a logarithm with base b.

Q: How do I solve a logarithmic equation with a variable in the exponent?

A: To solve a logarithmic equation with a variable in the exponent, we can use the property of logarithms that states:

$\log _a(x^y) = y \log _a(x)$

This property allows us to bring the exponent down and simplify the equation.

Q: What is the difference between a one-to-one function and a many-to-one function?

A: A one-to-one function is a function that maps each input to a unique output, while a many-to-one function is a function that maps multiple inputs to the same output. In the context of logarithmic equations, a one-to-one function is essential to ensure that the solution is unique.

Q: How do I check for extraneous solutions in a logarithmic equation?

A: To check for extraneous solutions, we can substitute the solution back into the original equation and verify that it is true. If the solution does not satisfy the original equation, it is an extraneous solution and should be discarded.

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

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

• Not applying the product property of logarithms
• Not simplifying the equation properly
• Not checking for extraneous solutions
• Not using the correct base
• Not using the change-of-base formula when necessary

Q: How do I apply the product property of logarithms to simplify a logarithmic equation?

A: To apply the product property of logarithms, we can use the formula:

$\log _a(b) + \log _a(c) = \log _a(bc)$

This formula allows us to combine two logarithms with the same base into a single logarithm.

Q: What is the importance of understanding logarithmic properties?

A: Understanding logarithmic properties is essential to solving logarithmic equations. Logarithmic properties provide a set of rules that allow us to manipulate logarithmic expressions and simplify equations. By understanding these properties, we can solve logarithmic equations more efficiently and accurately.

Q: How do I apply the change-of-base formula to convert a logarithm to a common or natural logarithm?

A: To apply the change-of-base formula, we can use the formula:

$\log _a(x) = \frac{\log _b(x)}{\log _b(a)}$

This formula allows us to convert a logarithm with base a to a logarithm with base b.

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

A: A logarithmic equation is an equation that involves a logarithm, while a polynomial equation is an equation that involves a polynomial expression. Logarithmic equations and polynomial equations have different properties and require different solution techniques.

Q: How do I solve a logarithmic equation with a variable in the argument?

A: To solve a logarithmic equation with a variable in the argument, we can use the property of logarithms that states:

$\log _a(x) = y \implies x = a^y$

This property allows us to bring the variable down and simplify the equation.