Which Of The Following Shows The Extraneous Solution To The Logarithmic Equation Log ⁡ 7 ( 3 X 3 + X ) − Log ⁡ 7 ( X ) = 2 \log_7(3x^3 + X) - \log_7(x) = 2 Lo G 7 ​ ( 3 X 3 + X ) − Lo G 7 ​ ( X ) = 2 ?A. X = − 16 X = -16 X = − 16 B. X = − 4 X = -4 X = − 4 C. X = 4 X = 4 X = 4 D. X = 16 X = 16 X = 16

Introduction

Logarithmic equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of logarithms. In this article, we will focus on solving the logarithmic equation $\log_7(3x^3 + x) - \log_7(x) = 2$ and identifying the extraneous solution among the given options.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithmic function. The general form of a logarithmic equation is $\log_b(x) = y$, where $b$ is the base of the logarithm, $x$ is the argument of the logarithm, and $y$ is the result of the logarithm.

Properties of Logarithms

To solve logarithmic equations, we need 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: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$

Solving the Logarithmic Equation

Now, let's solve the logarithmic equation $\log_7(3x^3 + x) - \log_7(x) = 2$.

First, we can use the quotient property to simplify the equation:

$$\log_7(3x^3 + x) - \log_7(x) = 2$$

$$\log_7\left(\frac{3x^3 + x}{x}\right) = 2$$

$$\log_7(3x^2 + 1) = 2$$

Next, we can use the definition of a logarithm to rewrite the equation:

$$7^2 = 3x^2 + 1$$

$$49 = 3x^2 + 1$$

Subtracting 1 from both sides gives:

$$48 = 3x^2$$

Dividing both sides by 3 gives:

$$16 = x^2$$

Taking the square root of both sides gives:

$$x = \pm 4$$

Identifying the Extraneous Solution

Now, let's identify the extraneous solution among the given options.

The given options are:

A. $x = -16$ B. $x = -4$ C. $x = 4$ D. $x = 16$

We can see that the correct solutions are $x = 4$ and $x = -4$. However, we need to check if these solutions are extraneous.

To check if a solution is extraneous, we need to plug it back into the original equation and check if it satisfies the equation.

Let's plug $x = -4$ back into the original equation:

$$\log_7(3(-4)^3 + (-4)) - \log_7(-4) = 2$$

$$\log_7(-192) - \log_7(-4) = 2$$

This equation is undefined because the logarithm of a negative number is undefined.

Therefore, $x = -4$ is an extraneous solution.

On the other hand, let's plug $x = 4$ back into the original equation:

$$\log_7(3(4)^3 + 4) - \log_7(4) = 2$$

$$\log_7(512) - \log_7(4) = 2$$

$$\log_7\left(\frac{512}{4}\right) = 2$$

$$\log_7(128) = 2$$

This equation is true, so $x = 4$ is not an extraneous solution.

Conclusion

In conclusion, the extraneous solution to the logarithmic equation $\log_7(3x^3 + x) - \log_7(x) = 2$ is $x = -4$.

Final Answer

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Introduction

Logarithmic equations can be challenging to solve, but with the right approach and understanding of the properties of logarithms, they can be tackled with ease. In this article, we will provide a Q&A guide to help you understand and solve logarithmic equations.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithmic function. The general form of a logarithmic equation is $\log_b(x) = y$, where $b$ is the base of the logarithm, $x$ is the argument of the logarithm, and $y$ is the result of the logarithm.

Q: What are the properties of logarithms?

A: The two main properties of logarithms are:

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

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can follow these steps:

1. Simplify the equation using the properties of logarithms.
2. Use the definition of a logarithm to rewrite the equation.
3. Solve for the variable.

Q: What is an extraneous solution?

A: An extraneous solution is a solution that is not valid for the original equation. It is a solution that is obtained by mistake or by using an incorrect method.

Q: How do I identify an extraneous solution?

A: To identify an extraneous solution, you can plug the solution back into the original equation and check if it satisfies the equation. If the solution does not satisfy the equation, it is an extraneous solution.

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. For example, $\log_7(x) = 2$ is a logarithmic equation, while $7^2 = x$ is an exponential equation.

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

A: Yes, here is an example of a logarithmic equation:

$$\log_7(3x^3 + x) - \log_7(x) = 2$$

This equation can be solved using the properties of logarithms and the definition of a logarithm.

Q: Can you provide an example of an extraneous solution?

A: Yes, here is an example of an extraneous solution:

$$\log_7(-4) = 2$$

This equation is undefined because the logarithm of a negative number is undefined. Therefore, $x = -4$ is an extraneous solution.

Conclusion

In conclusion, logarithmic equations can be challenging to solve, but with the right approach and understanding of the properties of logarithms, they can be tackled with ease. By following the steps outlined in this article, you can solve logarithmic equations and identify extraneous solutions.

Final Tips

• Make sure to understand the properties of logarithms before attempting to solve a logarithmic equation.
• Use the definition of a logarithm to rewrite the equation.
• Check for extraneous solutions by plugging the solution back into the original equation.

Common Mistakes

• Failing to simplify the equation using the properties of logarithms.
• Not using the definition of a logarithm to rewrite the equation.
• Not checking for extraneous solutions.

Additional Resources

• Khan Academy: Logarithmic Equations
• Mathway: Logarithmic Equations
• Wolfram Alpha: Logarithmic Equations