Which Statement About The Potential Solutions To $2 \log X - \log 3 = \log 3$ Is True?A. Both Are Extraneous Solutions. B. Only 3 Is An Extraneous Solution. C. Only -3 Is An Extraneous Solution. D. Neither Is An Extraneous Solution.

Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithmic terms. In this article, we will explore the potential solutions to the equation $2 \log x - \log 3 = \log 3$ and determine which statement about the solutions is true.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's review the properties of logarithmic equations. The logarithmic function is the inverse of the exponential function, and it is defined as:

$$\log_b a = c \iff b^c = a$$

where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Solving the Equation

To solve the equation $2 \log x - \log 3 = \log 3$, we can start by using the properties of logarithms to simplify the equation.

Using the property of logarithms that states $\log_b a - \log_b c = \log_b \frac{a}{c}$, we can rewrite the equation as:

$$\log x^2 - \log 3 = \log 3$$

Using the property of logarithms that states $\log_b a - \log_b c = \log_b \frac{a}{c}$, we can rewrite the equation as:

$$\log \frac{x^2}{3} = \log 3$$

Since the logarithmic function is one-to-one, we can equate the arguments of the logarithms:

$$\frac{x^2}{3} = 3$$

Multiplying both sides of the equation by 3, we get:

$$x^2 = 9$$

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

$$x = \pm 3$$

Analyzing the Solutions

Now that we have found the solutions to the equation, let's analyze them to determine which statement about the solutions is true.

The solutions to the equation are $x = 3$ and $x = -3$. To determine which statement is true, let's examine each option:

• Option A: Both are extraneous solutions. This option suggests that both $x = 3$ and $x = -3$ are extraneous solutions, meaning that they do not satisfy the original equation.
• Option B: Only 3 is an extraneous solution. This option suggests that only $x = 3$ is an extraneous solution, meaning that it does not satisfy the original equation.
• Option C: Only -3 is an extraneous solution. This option suggests that only $x = -3$ is an extraneous solution, meaning that it does not satisfy the original equation.
• Option D: Neither is an extraneous solution. This option suggests that neither $x = 3$ nor $x = -3$ is an extraneous solution, meaning that they both satisfy the original equation.

Conclusion

To determine which statement is true, let's substitute each solution back into the original equation and check if it is true.

Substituting $x = 3$ into the original equation, we get:

$$2 \log 3 - \log 3 = \log 3$$

Using the property of logarithms that states $\log_b a - \log_b c = \log_b \frac{a}{c}$, we can rewrite the equation as:

$$\log \frac{3^2}{3} = \log 3$$

Simplifying the equation, we get:

$$\log 3 = \log 3$$

This equation is true, so $x = 3$ is not an extraneous solution.

Substituting $x = -3$ into the original equation, we get:

$$2 \log (-3) - \log 3 = \log 3$$

Using the property of logarithms that states $\log_b a - \log_b c = \log_b \frac{a}{c}$, we can rewrite the equation as:

$$\log \frac{(-3)^2}{3} = \log 3$$

Simplifying the equation, we get:

$$\log 3 = \log 3$$

This equation is true, so $x = -3$ is not an extraneous solution.

However, we must consider the domain of the logarithmic function. The logarithmic function is defined only for positive real numbers, so $x = -3$ is not in the domain of the logarithmic function.

Therefore, $x = -3$ is an extraneous solution.

Conclusion

In conclusion, the statement that is true is:

• Option C: Only -3 is an extraneous solution.

This statement is true because $x = -3$ is an extraneous solution, while $x = 3$ is not.

Final Answer

Introduction

Logarithmic equations can be challenging to solve, especially when they involve multiple logarithmic terms. 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 logarithmic function is the inverse of the exponential function, and it is defined as:

$$\log_b a = c \iff b^c = a$$

where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithms to simplify the equation. Here are some common properties of logarithms:

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

You can use these properties to simplify the equation and isolate the variable.

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

A: A logarithmic equation and an exponential equation are related but distinct concepts.

• Logarithmic equation: A logarithmic equation is an equation that involves a logarithmic function. It is defined as:

$$\log_b a = c \iff b^c = a$$

where $b$ is the base of the logarithm, $a$ is the argument of the logarithm, and $c$ is the result of the logarithm.

• Exponential equation: An exponential equation is an equation that involves an exponential function. It is defined as:

$$b^c = a \iff \log_b a = c$$

where $b$ is the base of the exponential function, $a$ is the argument of the exponential function, and $c$ is the result of the exponential function.

Q: How do I determine the domain of a logarithmic function?

A: The domain of a logarithmic function is the set of all possible input values for which the function is defined. For a logarithmic function to be defined, the argument of the logarithm must be positive.

For example, the domain of the logarithmic function $\log x$ is all real numbers greater than 0, or $(0, \infty)$.

Q: What is the range of a logarithmic function?

A: The range of a logarithmic function is the set of all possible output values for which the function is defined. For a logarithmic function, the range is all real numbers.

For example, the range of the logarithmic function $\log x$ is all real numbers, or $(-\infty, \infty)$.

Q: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use the following steps:

1. Determine the domain: Determine the domain of the logarithmic function.
2. Determine the range: Determine the range of the logarithmic function.
3. Plot the points: Plot the points on the graph that correspond to the domain and range of the function.
4. Draw the graph: Draw the graph of the function using the points you plotted.

Q: What are some common applications of logarithmic equations?

A: Logarithmic equations have many applications in various fields, 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 an important concept in mathematics that have many applications in various fields. By understanding and solving logarithmic equations, you can gain a deeper understanding of the underlying mathematics and develop problem-solving skills.

Final Answer

The final answer is that logarithmic equations are a powerful tool for solving problems in various fields, and by understanding and solving them, you can gain a deeper understanding of the underlying mathematics and develop problem-solving skills.