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

Introduction

In this article, we will explore the solution to the logarithmic equation $\log _2(9x) - \log _2 3 = 3$. This equation involves logarithms with base 2 and requires the application of logarithmic properties to simplify and solve. We will break down the solution step by step, using the properties of logarithms to isolate the variable x.

Understanding Logarithmic Properties

Before diving into the solution, it's essential to understand the properties of logarithms. The logarithmic function is the inverse of the exponential function. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number. In this case, we are dealing with base 2 logarithms.

One of the key properties of logarithms is the product rule, which states that $\log_b (xy) = \log_b x + \log_b y$. This property allows us to combine the logarithms of two numbers into a single logarithm. Another important property is the quotient rule, which states that $\log_b \frac{x}{y} = \log_b x - \log_b y$. This property allows us to subtract the logarithm of one number from the logarithm of another.

Simplifying the Equation

Now that we have a basic understanding of logarithmic properties, let's apply them to simplify the given equation. We start by using the quotient rule to combine the two logarithms on the left-hand side of the equation:

$$\log _2(9x) - \log _2 3 = 3$$

Using the quotient rule, we can rewrite the equation as:

$$\log _2 \frac{9x}{3} = 3$$

Applying the Power Rule

Next, we can apply the power rule of logarithms, which states that $\log_b x^n = n \log_b x$. This property allows us to bring the exponent down as a coefficient. In this case, we can rewrite the equation as:

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

Isolating the Variable x

Now that we have simplified the equation, we can isolate the variable x. To do this, we can apply the definition of a logarithm, which states that if $\log_b x = y$, then $b^y = x$. In this case, we can rewrite the equation as:

$$2^3 = 3x$$

Solving for x

Finally, we can solve for x by dividing both sides of the equation by 3:

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

Evaluating the Expression

To evaluate the expression, we can calculate the value of $2^3$, which is equal to 8. Therefore, we can rewrite the expression as:

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

Conclusion

In this article, we have explored the solution to the logarithmic equation $\log _2(9x) - \log _2 3 = 3$. By applying the properties of logarithms, we were able to simplify the equation and isolate the variable x. The final solution is $x = \frac{8}{3}$.

Final Answer

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

Introduction

In our previous article, we explored the solution to the logarithmic equation $\log _2(9x) - \log _2 3 = 3$. In this article, we will address some of the frequently asked questions (FAQs) 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 is the exponent to which a base must be raised to produce a given number. In this case, we are dealing with base 2 logarithms.

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 property allows us to combine the logarithms of two numbers into a single logarithm.

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$. This property allows us to subtract the logarithm of one number from the logarithm of another.

Q: How do I apply the power rule of logarithms?

A: The power rule of logarithms states that $\log_b x^n = n \log_b x$. This property allows us to bring the exponent down as a coefficient. To apply the power rule, simply multiply the exponent by the logarithm of the base.

Q: How do I isolate the variable x in a logarithmic equation?

A: To isolate the variable x in a logarithmic equation, you can apply the definition of a logarithm, which states that if $\log_b x = y$, then $b^y = x$. This will allow you to rewrite the equation in exponential form and solve for x.

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

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

Q: Can I use a calculator to solve logarithmic equations?

A: Yes, you can use a calculator to solve logarithmic equations. However, it's essential to understand the properties of logarithms and how to apply them to simplify the equation before using a calculator.

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 properties of logarithms
• Not isolating the variable x correctly
• Not checking the domain of the logarithmic function
• Not considering the possibility of extraneous solutions

Conclusion

In this article, we have addressed some of the frequently asked questions (FAQs) about the solution to the logarithmic equation $\log _2(9x) - \log _2 3 = 3$. We hope that this article has provided you with a better understanding of the solution to this equation and has helped you to avoid common mistakes when solving logarithmic equations.

Final Answer

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