Simplify The Logarithmic Expression:$\[ \log_3(x+6) - \log_3(x-2) = 4 \\]This Can Be Rewritten As:$\[ \log_3\left(\frac{x+6}{x-2}\right) = 4 \\]

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Introduction

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Logarithmic expressions can be complex and challenging to simplify. However, with the right approach and techniques, we can rewrite them in a more manageable form. In this article, we will focus on simplifying the logarithmic expression $\log_3(x+6) - \log_3(x-2) = 4$. We will break down the problem step by step, using logarithmic properties and techniques to arrive at a simplified expression.

Understanding Logarithmic Properties

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Before we dive into the problem, let's review some essential logarithmic properties that we will use to simplify the expression.

• Product Property: $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$
• Quotient Property: $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$
• Power Property: $\log_b(x^y) = y \cdot \log_b(x)$

These properties will be instrumental in simplifying the given logarithmic expression.

Simplifying the Logarithmic Expression

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Now that we have reviewed the logarithmic properties, let's apply them to simplify the expression.

$\log_3(x+6) - \log_3(x-2) = 4$

Using the Quotient Property, we can rewrite the expression as:

$\log_3\left(\frac{x+6}{x-2}\right) = 4$

This is a significant step in simplifying the expression, as we have combined the two logarithms into a single logarithm.

Applying Exponentiation

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The next step is to apply exponentiation to both sides of the equation. Since the base of the logarithm is 3, we can raise 3 to the power of both sides to eliminate the logarithm.

$3^{\log_3\left(\frac{x+6}{x-2}\right)} = 3^4$

Using the Power Property, we can simplify the left-hand side of the equation:

$\frac{x+6}{x-2} = 3^4$

Solving for x

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Now that we have simplified the expression, we can solve for x. We can start by evaluating the right-hand side of the equation:

$3^4 = 81$

So, we have:

$\frac{x+6}{x-2} = 81$

To solve for x, we can cross-multiply:

$x+6 = 81(x-2)$

Expanding the right-hand side of the equation, we get:

$x+6 = 81x - 162$

Subtracting x from both sides, we get:

$6 = 80x - 162$

Adding 162 to both sides, we get:

$168 = 80x$

Dividing both sides by 80, we get:

$x = \frac{168}{80}$

Simplifying the fraction, we get:

$x = \frac{21}{10}$

Conclusion

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In this article, we simplified the logarithmic expression $\log_3(x+6) - \log_3(x-2) = 4$ using logarithmic properties and techniques. We applied exponentiation to both sides of the equation, eliminated the logarithm, and solved for x. The final solution is $x = \frac{21}{10}$. This demonstrates the power of logarithmic properties and techniques in simplifying complex expressions.

Frequently Asked Questions

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Q: What is the logarithmic expression $\log_3(x+6) - \log_3(x-2) = 4$?

A: This is a logarithmic expression that can be simplified using logarithmic properties and techniques.

Q: How do we simplify the logarithmic expression?

A: We can simplify the expression by applying the Quotient Property, exponentiation, and solving for x.

Q: What is the final solution to the logarithmic expression?

A: The final solution is $x = \frac{21}{10}$.

Q: What are the logarithmic properties used in this article?

A: The logarithmic properties used in this article are the Product Property, Quotient Property, and Power Property.

Q: How do we apply exponentiation to both sides of the equation?

A: We can apply exponentiation to both sides of the equation by raising the base of the logarithm to the power of both sides.

Q: How do we solve for x?

A: We can solve for x by cross-multiplying, expanding the right-hand side of the equation, and simplifying the resulting expression.

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Introduction

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In our previous article, we simplified the logarithmic expression $\log_3(x+6) - \log_3(x-2) = 4$ using logarithmic properties and techniques. However, we understand that there may be more questions and concerns regarding this topic. In this article, we will address some of the frequently asked questions and provide additional information to help you better understand logarithmic expression simplification.

Q&A: Logarithmic Expression Simplification

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Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. An exponential expression, on the other hand, involves an exponent, which is a power to which a number is raised.

Q: How do I know when to use the Quotient Property and when to use the Product Property?

A: The Quotient Property is used when you have a division operation inside the logarithm, while the Product Property is used when you have a multiplication operation inside the logarithm. For example, $\log_3(\frac{x+6}{x-2})$ would use the Quotient Property, while $\log_3(x+6) \cdot \log_3(x-2)$ would use the Product Property.

Q: Can I simplify a logarithmic expression with a base other than 10 or e?

A: Yes, you can simplify a logarithmic expression with any base. The properties of logarithms, such as the Quotient Property and the Product Property, still apply.

Q: How do I apply exponentiation to both sides of the equation?

A: To apply exponentiation to both sides of the equation, you need to raise the base of the logarithm to the power of both sides. For example, if you have $\log_3(x+6) - \log_3(x-2) = 4$, you would raise 3 to the power of both sides to get $3^{\log_3(x+6) - \log_3(x-2)} = 3^4$.

Q: What is the difference between a logarithmic equation and a logarithmic expression?

A: A logarithmic equation is an equation that involves a logarithm, while a logarithmic expression is an expression that involves a logarithm. A logarithmic equation typically has an equal sign (=) between the two expressions, while a logarithmic expression does not.

Q: Can I simplify a logarithmic expression with a negative exponent?

A: Yes, you can simplify a logarithmic expression with a negative exponent. To do this, you need to use the property of logarithms that states $\log_b(x^{-n}) = -n \cdot \log_b(x)$.

Q: How do I solve for x in a logarithmic equation?

A: To solve for x in a logarithmic equation, you need to isolate the variable x on one side of the equation. This typically involves using the properties of logarithms, such as the Quotient Property and the Product Property, to simplify the equation.

Additional Tips and Tricks

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Tip 1: Use the properties of logarithms to simplify the equation before solving for x.

Tip 2: Make sure to check your work by plugging the solution back into the original equation.

Tip 3: Use a calculator to check your work and ensure that your solution is accurate.

Tip 4: Practice, practice, practice! The more you practice simplifying logarithmic expressions and solving logarithmic equations, the more comfortable you will become with the process.

Conclusion

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In this article, we addressed some of the frequently asked questions and provided additional information to help you better understand logarithmic expression simplification. We hope that this article has been helpful in clarifying any confusion and providing you with the tools and techniques you need to simplify logarithmic expressions and solve logarithmic equations.

Frequently Asked Questions

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Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. An exponential expression, on the other hand, involves an exponent, which is a power to which a number is raised.

Q: How do I know when to use the Quotient Property and when to use the Product Property?

A: The Quotient Property is used when you have a division operation inside the logarithm, while the Product Property is used when you have a multiplication operation inside the logarithm.

Q: Can I simplify a logarithmic expression with a base other than 10 or e?

A: Yes, you can simplify a logarithmic expression with any base. The properties of logarithms, such as the Quotient Property and the Product Property, still apply.

Q: How do I apply exponentiation to both sides of the equation?

A: To apply exponentiation to both sides of the equation, you need to raise the base of the logarithm to the power of both sides.

Q: What is the difference between a logarithmic equation and a logarithmic expression?

A: A logarithmic equation is an equation that involves a logarithm, while a logarithmic expression is an expression that involves a logarithm.

Q: Can I simplify a logarithmic expression with a negative exponent?

A: Yes, you can simplify a logarithmic expression with a negative exponent. To do this, you need to use the property of logarithms that states $\log_b(x^{-n}) = -n \cdot \log_b(x)$.

Q: How do I solve for x in a logarithmic equation?

A: To solve for x in a logarithmic equation, you need to isolate the variable x on one side of the equation. This typically involves using the properties of logarithms, such as the Quotient Property and the Product Property, to simplify the equation.