Solve The Equation. Write The Solution Set With The Exact Solutions. Log ⁡ 3 ( N − 5 ) + Log ⁡ 3 ( N + 3 ) = 2 \log_3(n-5) + \log_3(n+3) = 2 Lo G 3 ​ ( N − 5 ) + Lo G 3 ​ ( N + 3 ) = 2 If There Is More Than One Solution, Separate The Answers With Commas.The Exact Solution Set Is □ \square □ .

Introduction

In this article, we will solve a logarithmic equation involving the sum of two logarithms with the same base. The equation is $\log_3(n-5) + \log_3(n+3) = 2$. Our goal is to find the exact solution set, which may contain multiple solutions.

Understanding Logarithmic Equations

Before we dive into solving the equation, let's briefly review the properties of logarithmic equations. A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The general form of a logarithmic equation is $\log_b(x) = y$, where $b$ is the base of the logarithm, $x$ is the argument of the logarithm, and $y$ is the result of the logarithm.

Solving the Equation

To solve the equation $\log_3(n-5) + \log_3(n+3) = 2$, we can use the property of logarithms that states $\log_b(x) + \log_b(y) = \log_b(xy)$. Applying this property to our equation, we get:

$$\log_3((n-5)(n+3)) = 2$$

Now, we can rewrite the equation in exponential form by raising the base of the logarithm to the power of the result:

$$3^2 = (n-5)(n+3)$$

Simplifying the equation, we get:

$$9 = n^2 - 2n - 15$$

Rearranging the equation to form a quadratic equation, we get:

$$n^2 - 2n - 24 = 0$$

Factoring the Quadratic Equation

To solve the quadratic equation $n^2 - 2n - 24 = 0$, we can factor it as:

$$(n - 6)(n + 4) = 0$$

Solving for n

Now, we can set each factor equal to zero and solve for $n$:

$$n - 6 = 0 \Rightarrow n = 6$$

$$n + 4 = 0 \Rightarrow n = -4$$

Checking the Solutions

Before we conclude that $n = 6$ and $n = -4$ are the solutions to the equation, we need to check if they satisfy the original equation. Plugging in $n = 6$ and $n = -4$ into the original equation, we get:

$$\log_3(6-5) + \log_3(6+3) = \log_3(1) + \log_3(9) = 0 + 2 = 2$$

$$\log_3(-4-5) + \log_3(-4+3) = \log_3(-9) + \log_3(-1)$$

Since $\log_3(-9)$ and $\log_3(-1)$ are undefined, we conclude that $n = -4$ is not a valid solution.

Conclusion

In conclusion, the exact solution set to the equation $\log_3(n-5) + \log_3(n+3) = 2$ is $n = \boxed{6}$.

Final Answer

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Introduction

In our previous article, we solved the logarithmic equation $\log_3(n-5) + \log_3(n+3) = 2$ and found the exact solution set to be $n = 6$. In this article, we will answer some frequently asked questions related to the solution of the equation.

Q: What is the base of the logarithm in the equation?

A: The base of the logarithm in the equation is 3.

Q: What is the property of logarithms used to solve the equation?

A: The property of logarithms used to solve the equation is $\log_b(x) + \log_b(y) = \log_b(xy)$.

Q: How do we rewrite the equation in exponential form?

A: We rewrite the equation in exponential form by raising the base of the logarithm to the power of the result. In this case, we have $3^2 = (n-5)(n+3)$.

Q: What is the quadratic equation formed after simplifying the equation?

A: The quadratic equation formed after simplifying the equation is $n^2 - 2n - 24 = 0$.

Q: How do we solve the quadratic equation?

A: We solve the quadratic equation by factoring it as $(n - 6)(n + 4) = 0$.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are $n = 6$ and $n = -4$.

Q: Why is $n = -4$ not a valid solution?

A: $n = -4$ is not a valid solution because it makes the argument of the logarithm negative, which is undefined.

Q: What is the final answer to the equation?

A: The final answer to the equation is $n = \boxed{6}$.

Frequently Asked Questions

• Q: What is the difference between a logarithmic equation and an exponential equation? A: A logarithmic equation is an equation that involves a logarithm, while an exponential equation is an equation that involves an exponent.
• Q: How do we solve a logarithmic equation? A: We solve a logarithmic equation by using the properties of logarithms and rewriting the equation in exponential form.
• Q: What is the base of the logarithm in a logarithmic equation? A: The base of the logarithm in a logarithmic equation is the number that is raised to the power of the result of the logarithm.

Conclusion

In conclusion, we have answered some frequently asked questions related to the solution of the logarithmic equation $\log_3(n-5) + \log_3(n+3) = 2$. We hope that this article has provided a clear understanding of the solution to the equation and has addressed any questions or concerns that you may have had.

Final Answer

The final answer is $n = \boxed{6}$.