Solve The Equation And Write The Solution Set With The Exact Solutions. Log ⁡ 3 ( 4 X − 13 ) = 1 + Log ⁡ 3 ( X + 1 \log_3(4x - 13) = 1 + \log_3(x + 1 Lo G 3 ​ ( 4 X − 13 ) = 1 + Lo G 3 ​ ( X + 1 ]The Exact Solution Set Is □ \square □ .

Introduction

Logarithmic equations can be challenging to solve, but with the right approach, they can be tackled with ease. In this article, we will focus on solving the equation $\log_3(4x - 13) = 1 + \log_3(x + 1)$ and finding the exact solution set. We will break down the solution into manageable steps, making it easy to follow and understand.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's take a moment to understand what logarithmic equations are. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b(x)$, then $b^y = x$. Logarithmic equations can be solved using various techniques, including the use of logarithmic properties and the change of base formula.

Step 1: Simplify the Equation

The first step in solving the equation is to simplify it by combining the logarithms on the right-hand side. We can do this using the property of logarithms that states $\log_b(x) + \log_b(y) = \log_b(xy)$. Applying this property to the equation, we get:

$$\log_3(4x - 13) = \log_3(x + 1) + 1$$

Step 2: Use the Change of Base Formula

The next step is to use the change of base formula to rewrite the equation in terms of a common logarithm. The change of base formula states that $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ is any positive real number. Applying this formula to the equation, we get:

$$\log_3(4x - 13) = \frac{\log(x + 1)}{\log(3)} + 1$$

Step 3: Simplify the Equation Further

Now that we have rewritten the equation in terms of a common logarithm, we can simplify it further by combining the logarithms on the right-hand side. We can do this using the property of logarithms that states $\log_b(x) + \log_b(y) = \log_b(xy)$. Applying this property to the equation, we get:

$$\log_3(4x - 13) = \log_3((x + 1) \cdot 3)$$

Step 4: Exponentiate Both Sides

The next step is to exponentiate both sides of the equation to get rid of the logarithms. We can do this by raising both sides of the equation to the power of 3, which is the base of the logarithm. Applying this step to the equation, we get:

$$4x - 13 = (x + 1) \cdot 3$$

Step 5: Solve for x

Now that we have simplified the equation, we can solve for x by isolating the variable on one side of the equation. We can do this by subtracting x from both sides of the equation and then dividing both sides by 3. Applying this step to the equation, we get:

$$4x - 13 = 3x + 3$$

$$4x - 3x = 3 + 13$$

$$x = 16$$

Conclusion

In this article, we have solved the logarithmic equation $\log_3(4x - 13) = 1 + \log_3(x + 1)$ and found the exact solution set. We have broken down the solution into manageable steps, making it easy to follow and understand. By using the properties of logarithms and the change of base formula, we have simplified the equation and solved for x. The final solution is x = 16.

The Exact Solution Set

The exact solution set is $\boxed{16}$.

Final Thoughts

Introduction

In our previous article, we solved the logarithmic equation $\log_3(4x - 13) = 1 + \log_3(x + 1)$ and found the exact solution set. In this article, we will provide a Q&A guide to help you understand the solution and answer any questions you may have.

Q: What is a logarithmic equation?

A: A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. In other words, if $y = \log_b(x)$, then $b^y = x$.

Q: How do I simplify a logarithmic equation?

A: To simplify a logarithmic equation, you can use the properties of logarithms, such as the product rule, the quotient rule, and the power rule. You can also use the change of base formula to rewrite the equation in terms of a common logarithm.

Q: What is the change of base formula?

A: The change of base formula is a formula that allows you to rewrite a logarithmic equation in terms of a common logarithm. The formula is $\log_b(x) = \frac{\log_a(x)}{\log_a(b)}$, where $a$ is any positive real number.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can follow these steps:

1. Simplify the equation using the properties of logarithms.
2. Use the change of base formula to rewrite the equation in terms of a common logarithm.
3. Exponentiate both sides of the equation to get rid of the logarithms.
4. Solve for x by isolating the variable on one side of the equation.

Q: What is the final solution to the equation $\log_3(4x - 13) = 1 + \log_3(x + 1)$?

A: The final solution to the equation is x = 16.

Q: Can you provide more examples of logarithmic equations?

A: Yes, here are a few more examples of logarithmic equations:

• $\log_2(x + 1) = 2 + \log_2(x - 1)$
• $\log_5(3x - 2) = 1 + \log_5(x + 4)$
• $\log_7(2x + 1) = 3 + \log_7(x - 2)$

Q: How do I know which base to use when solving a logarithmic equation?

A: When solving a logarithmic equation, you can use any base that is convenient. However, it is often easiest to use a base that is a power of 10, such as 2, 5, or 10.

Q: Can you provide more tips for solving logarithmic equations?

A: Yes, here are a few more tips for solving logarithmic equations:

• Make sure to simplify the equation before solving it.
• Use the change of base formula to rewrite the equation in terms of a common logarithm.
• Exponentiate both sides of the equation to get rid of the logarithms.
• Solve for x by isolating the variable on one side of the equation.

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the logarithmic equation $\log_3(4x - 13) = 1 + \log_3(x + 1)$. We have also provided more examples of logarithmic equations and tips for solving them. By following these steps and tips, you should be able to solve any logarithmic equation that comes your way.