What Is The Solution To The Equation Below?$\[ 3 \log _4 X = \log _4 32 + \log _4 2 \\]A. \[$ X = -8 \$\] B. \[$ X = -4 \$\] C. \[$ X = 4 \$\] D. \[$ X = 8 \$\]

Understanding the Equation

The given equation is $3 \log _4 x = \log _4 32 + \log _4 2$. To solve this equation, we need to apply the properties of logarithms and simplify the expression. The equation involves logarithms with base 4, and our goal is to find the value of x that satisfies the equation.

Applying the Properties of Logarithms

We can start by simplifying the right-hand side of the equation using the property of logarithms that states $\log _a b + \log _a c = \log _a (bc)$. Applying this property, we get:

$\log _4 32 + \log _4 2 = \log _4 (32 \cdot 2) = \log _4 64$

Simplifying the Equation

Now, we can rewrite the original equation as:

$3 \log _4 x = \log _4 64$

Using the Power Rule of Logarithms

We can use the power rule of logarithms, which states that $a \log _b x = \log _b x^a$. Applying this rule to the left-hand side of the equation, we get:

$\log _4 x^3 = \log _4 64$

Equating the Arguments

Since the logarithms have the same base, we can equate the arguments:

$x^3 = 64$

Solving for x

To solve for x, we can take the cube root of both sides of the equation:

$x = \sqrt[3]{64}$

Evaluating the Cube Root

The cube root of 64 is 4, so we have:

$x = 4$

Conclusion

Therefore, the solution to the equation $3 \log _4 x = \log _4 32 + \log _4 2$ is $x = 4$. This means that the value of x that satisfies the equation is 4.

Checking the Answer Choices

Let's check the answer choices to see if they match our solution:

A. $x = -8$ B. $x = -4$ C. $x = 4$ D. $x = 8$

Only answer choice C matches our solution, which is $x = 4$.

Final Answer

The final answer is C. $x = 4$.

Q: What is the equation $3 \log _4 x = \log _4 32 + \log _4 2$ trying to solve for?

A: The equation is trying to solve for the value of x.

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

A: The base of the logarithms in the equation is 4.

Q: How do we simplify the right-hand side of the equation?

A: We simplify the right-hand side of the equation by using the property of logarithms that states $\log _a b + \log _a c = \log _a (bc)$.

Q: What is the result of simplifying the right-hand side of the equation?

A: The result of simplifying the right-hand side of the equation is $\log _4 64$.

Q: How do we rewrite the original equation?

A: We rewrite the original equation as $3 \log _4 x = \log _4 64$.

Q: What is the power rule of logarithms, and how do we apply it to the equation?

A: The power rule of logarithms states that $a \log _b x = \log _b x^a$. We apply this rule to the left-hand side of the equation to get $\log _4 x^3 = \log _4 64$.

Q: How do we equate the arguments of the logarithms?

A: Since the logarithms have the same base, we can equate the arguments: $x^3 = 64$.

Q: How do we solve for x?

A: To solve for x, we take the cube root of both sides of the equation: $x = \sqrt[3]{64}$.

Q: What is the value of x?

A: The value of x is 4.

Q: How do we check the answer choices?

A: We check the answer choices to see if they match our solution.

Q: Which answer choice matches our solution?

A: Only answer choice C matches our solution, which is $x = 4$.

Q: What is the final answer to the equation?

A: The final answer to the equation is C. $x = 4$.

Q: What is the significance of the equation $3 \log _4 x = \log _4 32 + \log _4 2$?

A: The equation $3 \log _4 x = \log _4 32 + \log _4 2$ is a logarithmic equation that requires the application of logarithmic properties to solve for the value of x.

Q: What are some common logarithmic properties that are used to solve equations like $3 \log _4 x = \log _4 32 + \log _4 2$?

A: Some common logarithmic properties that are used to solve equations like $3 \log _4 x = \log _4 32 + \log _4 2$ include the product rule, the quotient rule, and the power rule.

Q: How do we apply the product rule of logarithms to the equation $3 \log _4 x = \log _4 32 + \log _4 2$?

A: We apply the product rule of logarithms to the equation $3 \log _4 x = \log _4 32 + \log _4 2$ by using the property that $\log _a b + \log _a c = \log _a (bc)$.

Q: What is the result of applying the product rule of logarithms to the equation $3 \log _4 x = \log _4 32 + \log _4 2$?

A: The result of applying the product rule of logarithms to the equation $3 \log _4 x = \log _4 32 + \log _4 2$ is $\log _4 64$.

Q: How do we apply the power rule of logarithms to the equation $3 \log _4 x = \log _4 64$?

A: We apply the power rule of logarithms to the equation $3 \log _4 x = \log _4 64$ by using the property that $a \log _b x = \log _b x^a$.

Q: What is the result of applying the power rule of logarithms to the equation $3 \log _4 x = \log _4 64$?

A: The result of applying the power rule of logarithms to the equation $3 \log _4 x = \log _4 64$ is $\log _4 x^3 = \log _4 64$.

Q: How do we equate the arguments of the logarithms in the equation $\log _4 x^3 = \log _4 64$?

A: We equate the arguments of the logarithms in the equation $\log _4 x^3 = \log _4 64$ by setting $x^3 = 64$.

Q: How do we solve for x in the equation $x^3 = 64$?

A: We solve for x in the equation $x^3 = 64$ by taking the cube root of both sides of the equation: $x = \sqrt[3]{64}$.

Q: What is the value of x in the equation $x = \sqrt[3]{64}$?

A: The value of x in the equation $x = \sqrt[3]{64}$ is 4.

Q: How do we check the answer choices in the equation $x = 4$?

A: We check the answer choices in the equation $x = 4$ by comparing the solution to the answer choices.

Q: Which answer choice matches the solution in the equation $x = 4$?

A: The answer choice that matches the solution in the equation $x = 4$ is C. $x = 4$.

Q: What is the final answer to the equation $3 \log _4 x = \log _4 32 + \log _4 2$?

A: The final answer to the equation $3 \log _4 x = \log _4 32 + \log _4 2$ is C. $x = 4$.