What Is The Solution To Log ⁡ 2 ( 9 X ) − Log ⁡ 2 3 = 3 \log _2(9 X)-\log _2 3=3 Lo G 2 ​ ( 9 X ) − Lo G 2 ​ 3 = 3 ?A. X = 3 8 X=\frac{3}{8} X = 8 3 ​ B. X = 8 3 X=\frac{8}{3} X = 3 8 ​ C. X = 3 X=3 X = 3 D. X = 9 X=9 X = 9

Introduction

In this article, we will delve into the world of logarithms and explore the solution to the equation $\log _2(9 x)-\log _2 3=3$. This equation involves logarithmic functions and requires a thorough understanding of their properties to solve. We will break down the solution step by step, using the properties of logarithms to simplify the equation and isolate the variable $x$.

Understanding Logarithmic Functions

Before we dive into the solution, let's take a moment to understand the concept of logarithmic functions. A logarithmic function is the inverse of an exponential function. In other words, if $y = 2^x$, then $x = \log_2 y$. The logarithmic function with base $b$ is denoted as $\log_b x$, and it represents the exponent to which the base $b$ must be raised to produce the number $x$.

Applying Logarithmic Properties

To solve the equation $\log _2(9 x)-\log _2 3=3$, we can use the property of logarithms that states $\log_b (m/n) = \log_b m - \log_b n$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

Simplifying the Equation

Using the property mentioned above, we can rewrite the equation as $\log _2(9 x) - \log _2 3 = \log _2 (9x/3) = 3$. This simplifies the equation and allows us to work with a single logarithmic term.

Isolating the Variable $x$

Now that we have simplified the equation, we can isolate the variable $x$ by raising both sides of the equation to the power of 2. This will eliminate the logarithmic term and leave us with an exponential equation.

Solving for $x$

Raising both sides of the equation to the power of 2 gives us $9x/3 = 2^3$. We can simplify this equation by multiplying both sides by 3, which gives us $9x = 8$. Finally, we can solve for $x$ by dividing both sides of the equation by 9, which gives us $x = 8/9$.

Conclusion

In conclusion, the solution to the equation $\log _2(9 x)-\log _2 3=3$ is $x = 8/9$. This solution was obtained by applying the properties of logarithmic functions, simplifying the equation, and isolating the variable $x$. We hope that this article has provided a clear and concise explanation of the solution to this equation.

Final Answer

The final answer is: $\boxed{\frac{8}{9}}$

Introduction

In our previous article, we explored the solution to the equation $\log _2(9 x)-\log _2 3=3$. This equation involves logarithmic functions and requires a thorough understanding of their properties to solve. In this article, we will address some of the most frequently asked questions (FAQs) about the solution to this equation.

Q: What is the definition of a logarithmic function?

A: A logarithmic function is the inverse of an exponential function. In other words, if $y = 2^x$, then $x = \log_2 y$. The logarithmic function with base $b$ is denoted as $\log_b x$, and it represents the exponent to which the base $b$ must be raised to produce the number $x$.

Q: What is the property of logarithms that allows us to combine two logarithmic terms into a single logarithmic term?

A: The property of logarithms that allows us to combine two logarithmic terms into a single logarithmic term is $\log_b (m/n) = \log_b m - \log_b n$. This property allows us to simplify the equation $\log _2(9 x)-\log _2 3=3$ by combining the two logarithmic terms on the left-hand side of the equation into a single logarithmic term.

Q: How do we isolate the variable $x$ in the equation $\log _2(9 x)-\log _2 3=3$?

A: We isolate the variable $x$ in the equation $\log _2(9 x)-\log _2 3=3$ by raising both sides of the equation to the power of 2. This eliminates the logarithmic term and leaves us with an exponential equation.

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 $x = 8/9$. This solution was obtained by applying the properties of logarithmic functions, simplifying the equation, and isolating the variable $x$.

Q: Can you provide a step-by-step solution to the equation $\log _2(9 x)-\log _2 3=3$?

A: Here is a step-by-step solution to the equation $\log _2(9 x)-\log _2 3=3$:

1. Apply the property of logarithms that allows us to combine two logarithmic terms into a single logarithmic term: $\log _2(9 x) - \log _2 3 = \log _2 (9x/3) = 3$.
2. Raise both sides of the equation to the power of 2: $9x/3 = 2^3$.
3. Simplify the equation by multiplying both sides by 3: $9x = 8$.
4. Solve for $x$ by dividing both sides of the equation by 9: $x = 8/9$.

Conclusion

In conclusion, we hope that this article has provided a clear and concise explanation of the solution to the equation $\log _2(9 x)-\log _2 3=3$. We have addressed some of the most frequently asked questions (FAQs) about the solution to this equation and provided a step-by-step solution to the equation.

Final Answer

The final answer is: $\boxed{\frac{8}{9}}$