Justify Each Step Of Solving The Equation In Tristen's Work.1. Write The Original Equation: ${ \log _3(x+1)-\log _3(x-6) =\log _3(2 X+2) } 2. U S E T H E P R O P E R T Y O F L O G A R I T H M S T H A T S T A T E S \[ 2. Use The Property Of Logarithms That States \[ 2. U Se T H E P Ro P Er T Yo F L O G A R I T Hm S T Ha T S T A T Es \[ \log_b(a) - \log_b(b) =

Introduction

In this article, we will delve into the world of logarithmic equations and explore the step-by-step process of solving a specific equation presented by Tristen. The equation in question is $\log _3(x+1)-\log _3(x-6) =\log _3(2 x+2)$. We will break down each step of the solution, justifying our actions and providing a clear understanding of the mathematical concepts involved.

Step 1: Write the Original Equation

The original equation is given as $\log _3(x+1)-\log _3(x-6) =\log _3(2 x+2)$. This equation involves logarithms with base 3 and a variable x. Our goal is to solve for x.

Step 2: Apply the Property of Logarithms

We will use the property of logarithms that states $\log_b(a) - \log_b(b) = \log_b(\frac{a}{b})$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation.

\log _3(x+1)-\log _3(x-6) = \log _3\left(\frac{x+1}{x-6}\right)

Step 3: Simplify the Equation

Now that we have combined the logarithmic terms, we can simplify the equation by equating the arguments of the logarithms.

\log _3\left(\frac{x+1}{x-6}\right) = \log _3(2 x+2)

Since the logarithms have the same base, we can drop the logarithms and equate the arguments.

\frac{x+1}{x-6} = 2 x+2

Step 4: Cross-Multiply

To eliminate the fraction, we will cross-multiply.

(x+1)(2 x+2) = (x-6)(2 x+2)

Step 5: Expand and Simplify

We will expand and simplify both sides of the equation.

2 x^2 + 2 x + 2 x + 2 = 2 x^2 - 12 x + 2 x + 12

Combining like terms, we get:

2 x^2 + 4 x + 2 = 2 x^2 - 10 x + 12

Step 6: Subtract 2x^2 from Both Sides

To eliminate the quadratic term, we will subtract 2x^2 from both sides of the equation.

4 x + 2 = -10 x + 12

Step 7: Add 10x to Both Sides

To isolate the variable x, we will add 10x to both sides of the equation.

14 x + 2 = 12

Step 8: Subtract 2 from Both Sides

To further isolate the variable x, we will subtract 2 from both sides of the equation.

14 x = 10

Step 9: Divide Both Sides by 14

Finally, we will divide both sides of the equation by 14 to solve for x.

x = \frac{10}{14}

Conclusion

Introduction

In our previous article, we walked through the step-by-step process of solving a logarithmic equation presented by Tristen. In this article, we will address some common questions and concerns that readers may have regarding the solution.

Q: What is the property of logarithms used in this solution?

A: The property of logarithms used in this solution is $\log_b(a) - \log_b(b) = \log_b(\frac{a}{b})$. This property allows us to combine the two logarithmic terms on the left-hand side of the equation.

Q: Why can we drop the logarithms and equate the arguments?

A: Since the logarithms have the same base, we can drop the logarithms and equate the arguments. This is a fundamental property of logarithms, which states that if $\log_b(a) = \log_b(c)$, then $a = c$.

Q: What is the significance of cross-multiplying?

A: Cross-multiplying is a technique used to eliminate fractions in an equation. In this solution, we cross-multiply to eliminate the fraction and simplify the equation.

Q: Why do we need to expand and simplify both sides of the equation?

A: Expanding and simplifying both sides of the equation helps us to eliminate any unnecessary terms and make the equation easier to solve.

Q: What is the purpose of subtracting 2x^2 from both sides of the equation?

A: Subtracting 2x^2 from both sides of the equation helps us to eliminate the quadratic term and make the equation easier to solve.

Q: Why do we need to add 10x to both sides of the equation?

A: Adding 10x to both sides of the equation helps us to isolate the variable x and make the equation easier to solve.

Q: What is the significance of dividing both sides of the equation by 14?

A: Dividing both sides of the equation by 14 is the final step in solving for the variable x. This step helps us to isolate x and find its value.

Q: What is the final solution to the equation?

A: The final solution to the equation is x = 10/14, which can be further simplified to x = 5/7.

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 property of logarithms correctly
• Not simplifying the equation properly
• Not isolating the variable x correctly
• Not checking for extraneous solutions

Conclusion

In this article, we have addressed some common questions and concerns that readers may have regarding the solution to the logarithmic equation presented by Tristen. We hope that this Q&A guide has provided additional clarity and insight into the solution process.