Which Of The Following Shows The Extraneous Solution To The Logarithmic Equation?A. $x = -16$ B. $x = -4$ C. $x = 4$ D. $x = 16$

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. One of the key aspects of solving logarithmic equations is identifying extraneous solutions. In this article, we will explore what extraneous solutions are and how to identify them in logarithmic equations.

What are Extraneous Solutions?

Extraneous solutions are solutions to an equation that do not satisfy the original equation. In other words, they are solutions that are not valid or are not true. Extraneous solutions can arise when solving equations involving logarithms, and it is essential to identify them to ensure that the solution is correct.

Properties of Logarithms

Before we dive into solving logarithmic equations, it is essential to understand the properties of logarithms. The two main properties of logarithms are:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

These properties will be used to solve logarithmic equations and identify extraneous solutions.

Solving Logarithmic Equations

To solve logarithmic equations, we need to isolate the logarithmic term. We can do this by using the properties of logarithms and algebraic manipulations.

Example 1: Solving a Logarithmic Equation

Let's consider the equation $\log_2 (x + 2) = 3$. To solve this equation, we need to isolate the logarithmic term.

$\log_2 (x + 2) = 3$

Using the definition of logarithms, we can rewrite the equation as:

$2^3 = x + 2$

$8 = x + 2$

Subtracting 2 from both sides, we get:

$6 = x$

Therefore, the solution to the equation is $x = 6$.

Identifying Extraneous Solutions

Now that we have solved the equation, we need to check if the solution is valid. In this case, we need to check if $x = 6$ satisfies the original equation.

$\log_2 (x + 2) = 3$

Substituting $x = 6$ into the equation, we get:

$\log_2 (6 + 2) = 3$

$\log_2 (8) = 3$

Since $2^3 = 8$, the solution $x = 6$ is valid.

Example 2: Identifying an Extraneous Solution

Let's consider the equation $\log_2 (x - 2) = 3$. To solve this equation, we need to isolate the logarithmic term.

$\log_2 (x - 2) = 3$

Using the definition of logarithms, we can rewrite the equation as:

$2^3 = x - 2$

$8 = x - 2$

Adding 2 to both sides, we get:

$10 = x$

Therefore, the solution to the equation is $x = 10$.

However, we need to check if the solution is valid. In this case, we need to check if $x = 10$ satisfies the original equation.

$\log_2 (x - 2) = 3$

Substituting $x = 10$ into the equation, we get:

$\log_2 (10 - 2) = 3$

$\log_2 (8) = 3$

Since $2^3 = 8$, the solution $x = 10$ is valid.

However, let's consider another solution, $x = -16$. Substituting $x = -16$ into the equation, we get:

$\log_2 (-16 - 2) = 3$

$\log_2 (-18) = 3$

Since the logarithm of a negative number is undefined, the solution $x = -16$ is not valid.

Therefore, the extraneous solution to the equation is $x = -16$.

Conclusion

In conclusion, extraneous solutions are solutions to an equation that do not satisfy the original equation. Identifying extraneous solutions is essential when solving logarithmic equations. By using the properties of logarithms and algebraic manipulations, we can solve logarithmic equations and identify extraneous solutions.

Final Answer

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Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In our previous article, we explored what extraneous solutions are and how to identify them in logarithmic equations. In this article, we will provide a Q&A guide to help you better 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 is used to solve equations that involve exponential expressions.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the logarithmic term. You can do this by using the properties of logarithms and algebraic manipulations. The two main properties of logarithms are:

• Product Property: $\log_b (xy) = \log_b x + \log_b y$
• Quotient Property: $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$

Q: What is an extraneous solution?

A: An extraneous solution is a solution to an equation that does not satisfy the original equation. In other words, it is a solution that is not valid or is not true. Extraneous solutions can arise when solving equations involving logarithms, and it is essential to identify them to ensure that the solution is correct.

Q: How do I identify an extraneous solution?

A: To identify an extraneous solution, you need to check if the solution satisfies the original equation. You can do this by substituting the solution into the equation and checking if it is true.

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 for extraneous solutions: Failing to check for extraneous solutions can lead to incorrect solutions.
• Not using the correct properties of logarithms: Using the wrong properties of logarithms can lead to incorrect solutions.
• Not simplifying the equation: Failing to simplify the equation can make it difficult to solve.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you need to use the properties of logarithms and algebraic manipulations. You can do this by:

• Using the product property: Using the product property to combine logarithmic terms.
• Using the quotient property: Using the quotient property to divide logarithmic terms.
• Simplifying the equation: Simplifying the equation by combining like terms.

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

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

• Finance: Logarithmic equations are used to calculate interest rates and investment returns.
• Science: Logarithmic equations are used to model population growth and decay.
• Engineering: Logarithmic equations are used to design and optimize systems.

Conclusion

In conclusion, logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. By following the steps outlined in this article, you can solve logarithmic equations and identify extraneous solutions. Remember to check for extraneous solutions and use the correct properties of logarithms to ensure that your solutions are correct.

Final Answer

The final answer is: A