What Is The Solution To $\log _2(9 X) - \log _2 3 = 3$?A. $x = \frac{3}{8}$ B. $x = \frac{8}{3}$ C. $x = 3$ D. $x = 9$

Introduction

Mathematical equations involving logarithms can be challenging to solve, especially when they involve variables within the logarithmic function. In this article, we will explore the solution to the equation $\log _2(9 x) - \log _2 3 = 3$. This equation involves logarithms with base 2 and a variable $x$ within the logarithmic function. Our goal is to isolate the variable $x$ and determine its value.

Understanding Logarithmic Properties

Before we dive into solving the equation, it's essential to understand some fundamental properties of logarithms. The logarithmic function $\log _a b$ represents the exponent to which the base $a$ must be raised to produce the number $b$. For example, $\log _2 8 = 3$ because $2^3 = 8$. The logarithmic function is the inverse of the exponential function.

One of the key properties of logarithms is the product rule, which states that $\log _a (b \cdot c) = \log _a b + \log _a c$. This property allows us to simplify logarithmic expressions by combining the logarithms of multiple factors.

Another important property is the quotient rule, which states that $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$. This property allows us to simplify logarithmic expressions by subtracting the logarithms of two factors.

Applying Logarithmic Properties to the Equation

Now that we have a solid understanding of logarithmic properties, let's apply them to the equation $\log _2(9 x) - \log _2 3 = 3$. Using the quotient rule, we can rewrite the equation as:

$$\log _2 \left(\frac{9 x}{3}\right) = 3$$

Simplifying the fraction inside the logarithmic function, we get:

$$\log _2 (3 x) = 3$$

Isolating the Variable $x$

Now that we have simplified the equation, let's isolate the variable $x$. To do this, we can use the definition of the logarithmic function. Since $\log _a b = c$ is equivalent to $a^c = b$, we can rewrite the equation as:

$$2^3 = 3 x$$

Simplifying the left-hand side, we get:

$$8 = 3 x$$

Solving for $x$

Now that we have isolated the variable $x$, let's solve for its value. To do this, we can divide both sides of the equation by 3:

$$x = \frac{8}{3}$$

Conclusion

In this article, we explored the solution to the equation $\log _2(9 x) - \log _2 3 = 3$. We applied logarithmic properties to simplify the equation and isolate the variable $x$. Our final solution is $x = \frac{8}{3}$.

Discussion

The solution to the equation $\log _2(9 x) - \log _2 3 = 3$ is $x = \frac{8}{3}$. This solution can be verified by plugging it back into the original equation. If you have any questions or would like to discuss this solution further, please feel free to leave a comment below.

Final Answer

The final answer is: $\boxed{\frac{8}{3}}$

Introduction

In our previous article, we explored the solution to the equation $\log _2(9 x) - \log _2 3 = 3$. We applied logarithmic properties to simplify the equation and isolate the variable $x$. Our final solution was $x = \frac{8}{3}$. In this article, we will answer some frequently asked questions about the solution to this equation.

Q: What is the definition of a logarithm?

A: A logarithm is the inverse of an exponential function. It represents the exponent to which the base must be raised to produce the number. For example, $\log _2 8 = 3$ because $2^3 = 8$.

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log _a (b \cdot c) = \log _a b + \log _a c$. This property allows us to simplify logarithmic expressions by combining the logarithms of multiple factors.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log _a \left(\frac{b}{c}\right) = \log _a b - \log _a c$. This property allows us to simplify logarithmic expressions by subtracting the logarithms of two factors.

Q: How did you simplify the equation $\log _2(9 x) - \log _2 3 = 3$?

A: We applied the quotient rule of logarithms to simplify the equation. We rewrote the equation as $\log _2 \left(\frac{9 x}{3}\right) = 3$ and then simplified the fraction inside the logarithmic function to get $\log _2 (3 x) = 3$.

Q: How did you isolate the variable $x$?

A: We used the definition of the logarithmic function to rewrite the equation as $2^3 = 3 x$. We then simplified the left-hand side to get $8 = 3 x$ and finally solved for $x$ by dividing both sides of the equation by 3.

Q: What is the final solution to the equation $\log _2(9 x) - \log _2 3 = 3$?

A: The final solution to the equation $\log _2(9 x) - \log _2 3 = 3$ is $x = \frac{8}{3}$.

Q: How can I verify the solution?

A: You can verify the solution by plugging it back into the original equation. If the equation holds true, then the solution is correct.

Q: What are some common mistakes to avoid when solving logarithmic equations?

A: Some common mistakes to avoid when solving logarithmic equations include:

• Not applying the correct logarithmic properties
• Not simplifying the equation correctly
• Not isolating the variable correctly
• Not verifying the solution

Conclusion

In this article, we answered some frequently asked questions about the solution to the equation $\log _2(9 x) - \log _2 3 = 3$. We provided explanations and examples to help clarify the concepts and procedures involved in solving logarithmic equations. We hope this article has been helpful in understanding the solution to this equation.

Final Answer

The final answer is: $\boxed{\frac{8}{3}}$