Which Expression Is Equivalent To Log ⁡ 18 − Log ⁡ ( P + 2 \log 18 - \log (p+2 Lo G 18 − Lo G ( P + 2 ]?A. Log ⁡ P + 2 18 \log \frac{p+2}{18} Lo G 18 P + 2 ​ B. Log ⁡ 18 P + 2 \log \frac{18}{p+2} Lo G P + 2 18 ​ C. Log ⁡ 20 P \log \frac{20}{p} Lo G P 20 ​ D. $\log [18(p+2)]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will focus on simplifying the expression $\log 18 - \log (p+2)$ and explore the different options provided.

Understanding Logarithmic Properties

Before we dive into simplifying the expression, it's essential to understand the properties of logarithms. The two main properties of logarithms are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \frac{a}{b} = \log a - \log b$

These properties will be instrumental in simplifying the given expression.

Simplifying the Expression

The given expression is $\log 18 - \log (p+2)$. To simplify this expression, we can use the Quotient Property of logarithms, which states that $\log \frac{a}{b} = \log a - \log b$. In this case, we can rewrite the expression as:

$$\log 18 - \log (p+2) = \log \frac{18}{p+2}$$

This simplification is based on the Quotient Property, where we can combine the two logarithmic terms into a single logarithmic term with a fraction.

Evaluating the Options

Now that we have simplified the expression, let's evaluate the options provided:

• Option A: $\log \frac{p+2}{18}$ - This option is incorrect because it reverses the order of the numerator and denominator.
• Option B: $\log \frac{18}{p+2}$ - This option is correct because it matches the simplified expression we obtained earlier.
• Option C: $\log \frac{20}{p}$ - This option is incorrect because it introduces a new constant (20) and changes the denominator to $p$.
• Option D: $\log [18(p+2)]$ - This option is incorrect because it multiplies the two terms inside the logarithm instead of dividing them.

Conclusion

In conclusion, the correct expression equivalent to $\log 18 - \log (p+2)$ is $\log \frac{18}{p+2}$. This simplification is based on the Quotient Property of logarithms, which allows us to combine the two logarithmic terms into a single logarithmic term with a fraction. By understanding and applying logarithmic properties, we can simplify complex expressions and solve mathematical problems with ease.

Additional Tips and Tricks

Here are some additional tips and tricks to help you simplify logarithmic expressions:

• Use the Product Property: When simplifying expressions with multiple logarithmic terms, use the Product Property to combine them into a single logarithmic term.
• Use the Quotient Property: When simplifying expressions with two logarithmic terms, use the Quotient Property to combine them into a single logarithmic term with a fraction.
• Simplify the Expression: Before simplifying the expression, try to simplify the terms inside the logarithm by factoring or canceling out common factors.

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to simplify them.

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

A: A logarithmic expression is an expression that involves the logarithm of a number, while an exponential expression is an expression that involves the exponentiation of a number. For example, $\log 18$ is a logarithmic expression, while $2^3$ is an exponential expression.

Q: What are the two main properties of logarithms?

A: The two main properties of logarithms are:

• Product Property: $\log (ab) = \log a + \log b$
• Quotient Property: $\log \frac{a}{b} = \log a - \log b$

These properties will be instrumental in simplifying logarithmic expressions.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, use the Product Property to combine the terms into a single logarithmic term. For example, $\log 18 + \log (p+2)$ can be simplified as $\log (18(p+2))$.

Q: How do I simplify a logarithmic expression with two terms?

A: To simplify a logarithmic expression with two terms, use the Quotient Property to combine the terms into a single logarithmic term with a fraction. For example, $\log 18 - \log (p+2)$ can be simplified as $\log \frac{18}{p+2}$.

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

A: A logarithmic expression is an expression that involves the logarithm of a number, while an algebraic expression is an expression that involves variables and constants. For example, $\log 18$ is a logarithmic expression, while $2x+3$ is an algebraic expression.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to find the value of the expression inside the logarithm. For example, to evaluate $\log 18$, you need to find the value of $18$.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log 10$
• $\log 100$
• $\log 1000$
• $\log (p+2)$
• $\log (p-2)$

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding how to simplify them is crucial for solving various mathematical problems. By following the tips and tricks provided in this article, you can simplify logarithmic expressions with ease and become proficient in solving mathematical problems.

Additional Resources

Here are some additional resources to help you learn more about logarithmic expressions:

• Logarithmic Properties: A comprehensive guide to logarithmic properties, including the Product Property and the Quotient Property.
• Logarithmic Expressions: A tutorial on how to simplify logarithmic expressions, including examples and practice problems.
• Logarithmic Functions: A guide to logarithmic functions, including the definition, properties, and applications of logarithmic functions.

By following these resources, you can gain a deeper understanding of logarithmic expressions and become proficient in solving mathematical problems.