Which Of The Following Shows The Extraneous Solution To The Equation Log ⁡ 4 ( X ) + Log ⁡ 4 ( X − 3 ) = Log ⁡ 4 ( − 7 X + 21 \log_4(x) + \log_4(x-3) = \log_4(-7x + 21 Lo G 4 ​ ( X ) + Lo G 4 ​ ( X − 3 ) = Lo G 4 ​ ( − 7 X + 21 ]?A. X = − 7 X = -7 X = − 7 B. X = − 3 X = -3 X = − 3 C. X = 3 X = 3 X = 3 And X = − 7 X = -7 X = − 7 D. X = 7 X = 7 X = 7 And X = − 3 X = -3 X = − 3

Introduction

Logarithmic equations can be challenging to solve, especially when dealing with multiple logarithms. In this article, we will explore the concept of extraneous solutions in logarithmic equations and how to identify them. We will use the equation $\log_4(x) + \log_4(x-3) = \log_4(-7x + 21)$ as an example to demonstrate the process.

Understanding Logarithmic Equations

A logarithmic equation is an equation that contains a logarithm. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, $\log_4(16) = 2$ because $4^2 = 16$. Logarithmic equations can be solved using various techniques, including the use of logarithmic properties.

The Equation

The given equation is $\log_4(x) + \log_4(x-3) = \log_4(-7x + 21)$. To solve this equation, we can use the property of logarithms that states $\log_a(b) + \log_a(c) = \log_a(bc)$. Applying this property to the given equation, we get:

$$\log_4(x(x-3)) = \log_4(-7x + 21)$$

Simplifying the Equation

We can simplify the equation by eliminating the logarithms. Since the bases of the logarithms are the same, we can equate the expressions inside the logarithms:

$$x(x-3) = -7x + 21$$

Expanding the left-hand side of the equation, we get:

$$x^2 - 3x = -7x + 21$$

Rearranging the terms, we get:

$$x^2 + 4x - 21 = 0$$

Factoring the Quadratic Equation

The quadratic equation $x^2 + 4x - 21 = 0$ can be factored as:

$$(x + 7)(x - 3) = 0$$

Solving for x

To solve for x, we can set each factor equal to zero and solve for x:

$$x + 7 = 0 \Rightarrow x = -7$$

$$x - 3 = 0 \Rightarrow x = 3$$

Identifying Extraneous Solutions

Now that we have found the solutions to the equation, we need to check if they are extraneous. An extraneous solution is a solution that is not valid for the original equation. In this case, we need to check if the solutions $x = -7$ and $x = 3$ are valid.

Checking the Solutions

To check if the solutions are valid, we need to plug them back into the original equation and check if the equation holds true. Let's start with the solution $x = -7$:

$$\log_4(-7) + \log_4(-7-3) = \log_4(-7(-7) + 21)$$

$$\log_4(-7) + \log_4(-10) = \log_4(49 + 21)$$

$$\log_4(-7) + \log_4(-10) = \log_4(70)$$

Since the left-hand side of the equation is undefined (because the logarithm of a negative number is undefined), we can conclude that $x = -7$ is an extraneous solution.

Checking the Second Solution

Now let's check the second solution $x = 3$:

$$\log_4(3) + \log_4(3-3) = \log_4(-7(3) + 21)$$

$$\log_4(3) + \log_4(0) = \log_4(-21 + 21)$$

$$\log_4(3) + \log_4(0) = \log_4(0)$$

Since the left-hand side of the equation is undefined (because the logarithm of zero is undefined), we can conclude that $x = 3$ is also an extraneous solution.

Conclusion

In conclusion, the extraneous solution to the equation $\log_4(x) + \log_4(x-3) = \log_4(-7x + 21)$ is $x = -7$ and $x = 3$. These solutions are not valid for the original equation and are therefore extraneous.

Final Answer

Introduction

Logarithmic equations can be challenging to solve, especially when dealing with multiple logarithms. In this article, we will explore the concept of extraneous solutions in logarithmic equations and how to identify them. We will also provide a Q&A guide to help you better understand the topic.

Q&A Guide

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid for the original equation. In other words, it is a solution that does not satisfy the original equation.

Q: How do I identify extraneous solutions?

A: To identify extraneous solutions, you need to plug the solutions back into the original equation and check if the equation holds true. If the equation does not hold true, then the solution is extraneous.

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
• Not using the correct properties of logarithms
• Not simplifying the equation correctly
• Not factoring the quadratic equation correctly

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms, such as the product rule and the quotient rule. You can also use the fact that $\log_a(b) = c$ is equivalent to $a^c = b$.

Q: What is the product rule for logarithms?

A: The product rule for logarithms states that $\log_a(b) + \log_a(c) = \log_a(bc)$. This means that the logarithm of a product is equal to the sum of the logarithms of the factors.

Q: What is the quotient rule for logarithms?

A: The quotient rule for logarithms states that $\log_a(b) - \log_a(c) = \log_a(\frac{b}{c})$. This means that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use factoring, the quadratic formula, or completing the square. The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I check if a solution is valid?

A: To check if a solution is valid, you need to plug the solution back into the original equation and check if the equation holds true. If the equation does not hold true, then the solution is not valid.

Conclusion

In conclusion, solving logarithmic equations requires a good understanding of the properties of logarithms and the techniques for simplifying and solving quadratic equations. By following the steps outlined in this article and using the Q&A guide, you should be able to solve logarithmic equations with confidence.

Final Tips

• Always check for extraneous solutions
• Use the correct properties of logarithms
• Simplify the equation correctly
• Factor the quadratic equation correctly
• Check if the solution is valid by plugging it back into the original equation

Common Logarithmic Equations

• $\log_a(x) = y$
• $\log_a(x) + \log_a(y) = \log_a(xy)$
• $\log_a(x) - \log_a(y) = \log_a(\frac{x}{y})$
• $\log_a(x^y) = y\log_a(x)$

Logarithmic Properties

• $\log_a(1) = 0$
• $\log_a(a) = 1$
• $\log_a(b^c) = c\log_a(b)$
• $\log_a(\frac{b}{c}) = \log_a(b) - \log_a(c)$

Quadratic Formula

• $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Extraneous Solutions

• An extraneous solution is a solution that is not valid for the original equation.
• To identify extraneous solutions, plug the solutions back into the original equation and check if the equation holds true.