What Is The Solution To The Equation Below? Log ⁡ 6 4 X 2 − Log ⁡ 6 X = 2 \log_6 4x^2 - \log_6 X = 2 Lo G 6 ​ 4 X 2 − Lo G 6 ​ X = 2 A. X = 1 12 X = \frac{1}{12} X = 12 1 ​ B. X = 3 2 X = \frac{3}{2} X = 2 3 ​ C. X = 3 X = 3 X = 3 D. X = 9 X = 9 X = 9

Introduction

The given equation is a logarithmic equation that involves the base-6 logarithm. The equation is $\log_6 4x^2 - \log_6 x = 2$. To solve this equation, we need to apply the properties of logarithms and then isolate the variable $x$. In this article, we will walk through the step-by-step solution to the equation and provide the final answer.

Understanding the Properties of Logarithms

Before we dive into solving the equation, it's essential to understand the properties of logarithms. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In other words, if $y = \log_b x$, then $b^y = x$. This property is known as the definition of a logarithm.

Another important property of logarithms is the product rule, which states that $\log_b (xy) = \log_b x + \log_b y$. This property can be used to simplify expressions involving logarithms.

Applying the Properties of Logarithms to the Equation

Now that we have a good understanding of the properties of logarithms, let's apply them to the given equation. We can start by using the product rule to simplify the expression $\log_6 4x^2 - \log_6 x$.

Using the product rule, we can rewrite $\log_6 4x^2$ as $\log_6 4 + \log_6 x^2$. This is because $\log_6 4x^2 = \log_6 (4 \cdot x^2) = \log_6 4 + \log_6 x^2$.

Now, we can substitute this expression back into the original equation: $\log_6 4 + \log_6 x^2 - \log_6 x = 2$.

Simplifying the Equation

Next, we can simplify the equation by combining the logarithmic terms. Using the product rule again, we can rewrite $\log_6 x^2$ as $2\log_6 x$. This is because $\log_6 x^2 = \log_6 (x \cdot x) = \log_6 x + \log_6 x = 2\log_6 x$.

Now, we can substitute this expression back into the equation: $\log_6 4 + 2\log_6 x - \log_6 x = 2$.

Isolating the Variable $x$

Now that we have simplified the equation, we can isolate the variable $x$. We can start by combining the logarithmic terms: $\log_6 4 + \log_6 x = 2$.

Next, we can use the definition of a logarithm to rewrite the equation in exponential form: $6^2 = 4 \cdot x$.

Solving for $x$

Now that we have the equation in exponential form, we can solve for $x$. We can start by dividing both sides of the equation by 4: $x = \frac{6^2}{4}$.

Evaluating the Expression

Finally, we can evaluate the expression to find the value of $x$: $x = \frac{36}{4} = 9$.

Conclusion

In this article, we have walked through the step-by-step solution to the equation $\log_6 4x^2 - \log_6 x = 2$. We have applied the properties of logarithms, simplified the equation, and isolated the variable $x$. The final answer is $x = 9$.

Discussion

The solution to the equation is $x = 9$. This means that the value of $x$ that satisfies the equation is 9.

Final Answer

The final answer is $x = 9$.

Introduction

In the previous article, we solved the equation $\log_6 4x^2 - \log_6 x = 2$ and found that the solution is $x = 9$. However, we understand that readers may have questions about the equation and its solution. In this article, we will address some of the frequently asked questions (FAQs) about the equation.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is 6. This means that the logarithm is a base-6 logarithm.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log_b (xy) = \log_b x + \log_b y$. This means that the logarithm of a product can be rewritten as the sum of the logarithms of the individual factors.

Q: How do you simplify the expression $\log_6 4x^2 - \log_6 x$?

A: To simplify the expression, we can use the product rule to rewrite $\log_6 4x^2$ as $\log_6 4 + \log_6 x^2$. We can then use the definition of a logarithm to rewrite $\log_6 x^2$ as $2\log_6 x$. This gives us the simplified expression $\log_6 4 + 2\log_6 x - \log_6 x$.

Q: How do you isolate the variable $x$ in the equation?

A: To isolate the variable $x$, we can start by combining the logarithmic terms: $\log_6 4 + \log_6 x = 2$. We can then use the definition of a logarithm to rewrite the equation in exponential form: $6^2 = 4 \cdot x$. Finally, we can solve for $x$ by dividing both sides of the equation by 4.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = 9$.

Q: Why is the solution $x = 9$?

A: The solution $x = 9$ is the value of $x$ that satisfies the equation. In other words, when we substitute $x = 9$ into the equation, the equation is true.

Q: Can you provide more examples of logarithmic equations?

A: Yes, we can provide more examples of logarithmic equations. However, the solution to this particular equation is $x = 9$.

Q: How do you evaluate the expression $\frac{6^2}{4}$?

A: To evaluate the expression, we can start by calculating the value of $6^2$, which is 36. We can then divide 36 by 4 to get the final answer, which is 9.

Q: What is the significance of the base-6 logarithm in the equation?

A: The base-6 logarithm in the equation is used to simplify the expression and isolate the variable $x$. The base-6 logarithm is a common base used in logarithmic equations.

Q: Can you provide more information about logarithmic equations?

A: Yes, we can provide more information about logarithmic equations. However, the solution to this particular equation is $x = 9$.

Conclusion

In this article, we have addressed some of the frequently asked questions (FAQs) about the equation $\log_6 4x^2 - \log_6 x = 2$. We have provided explanations and examples to help readers understand the equation and its solution. The final answer to the equation is $x = 9$.