Which Expression Is Equivalent To $\log 18 - \log (p+2$\]?A. $\log \frac{p+2}{18}$ B. $\log \frac{18}{p+2}$ C. $\log \frac{20}{p}$ D. $\log [18 \cdot (p+2)\]

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Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will explore the concept of logarithmic expressions and how to simplify them using the properties of logarithms. We will also examine a specific problem that involves logarithmic expressions and determine which expression is equivalent to the given expression.

Properties of Logarithms

Before we dive into the problem, let's review the properties of logarithms. The logarithm of a number is the exponent to which a base number must be raised to produce that number. For example, the logarithm of 100 to the base 10 is 2, because 10^2 = 100.

There are several properties of logarithms that we need to know:

  • Product Property: log(a * b) = log(a) + log(b)
  • Quotient Property: log(a / b) = log(a) - log(b)
  • Power Property: log(a^b) = b * log(a)

The Problem

The problem we are given is to find an equivalent expression to log 18 - log (p+2). We need to simplify this expression using the properties of logarithms.

Step 1: Apply the Quotient Property

The first step is to apply the quotient property of logarithms, which states that log(a / b) = log(a) - log(b). In this case, we can rewrite the expression as:

log 18 - log (p+2) = log (18 / (p+2))

Step 2: Simplify the Expression

Now that we have applied the quotient property, we can simplify the expression further. We can rewrite the expression as:

log (18 / (p+2)) = log (18 / (p+2))

This is the simplified expression.

Comparing the Options

Now that we have simplified the expression, we can compare it to the options given:

A. log (p+2) / 18 B. log (18 / (p+2)) C. log (20 / p) D. log (18 * (p+2))

The only option that matches our simplified expression is option B: log (18 / (p+2)).

Conclusion

In this article, we explored the concept of logarithmic expressions and how to simplify them using the properties of logarithms. We also examined a specific problem that involved logarithmic expressions and determined which expression is equivalent to the given expression. The correct answer is option B: log (18 / (p+2)).

Final Answer

Introduction

In our previous article, we explored the concept of logarithmic expressions and how to simplify them using the properties of logarithms. We also examined a specific problem that involved logarithmic expressions and determined which expression is equivalent to the given expression. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and how to apply the properties of logarithms.

Q&A

Q: What is the product property of logarithms?

A: The product property of logarithms states that log(a * b) = log(a) + log(b). This means that the logarithm of a product is equal to the sum of the logarithms of the individual numbers.

Q: How do I apply the quotient property of logarithms?

A: To apply the quotient property of logarithms, you need to subtract the logarithm of the divisor from the logarithm of the dividend. For example, log(a / b) = log(a) - log(b).

Q: What is the power property of logarithms?

A: The power property of logarithms states that log(a^b) = b * log(a). This means that the logarithm of a number raised to a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I simplify a logarithmic expression using the properties of logarithms?

A: To simplify a logarithmic expression, you need to apply the properties of logarithms in the correct order. For example, if you have the expression log(a / b) + log(c), you can simplify it by applying the quotient property and then the product property.

Q: What is the difference between log(a) and log(b)?

A: The difference between log(a) and log(b) is the logarithm of the ratio of a and b. This can be expressed as log(a / b) = log(a) - log(b).

Q: How do I evaluate a logarithmic expression with a negative exponent?

A: To evaluate a logarithmic expression with a negative exponent, you need to apply the power property of logarithms. For example, log(a^(-b)) = -b * log(a).

Q: What is the relationship between logarithms and exponents?

A: Logarithms and exponents are inverse operations. This means that if you have a logarithmic expression, you can convert it to an exponential expression by applying the power property of logarithms.

Q: How do I use logarithmic expressions in real-world problems?

A: Logarithmic expressions are used in a variety of real-world problems, including finance, science, and engineering. For example, logarithmic expressions can be used to model population growth, chemical reactions, and financial transactions.

Conclusion

In this article, we provided a Q&A guide to help you better understand logarithmic expressions and how to apply the properties of logarithms. We hope that this guide has been helpful in clarifying any confusion you may have had about logarithmic expressions.

Final Tips

  • Always apply the properties of logarithms in the correct order.
  • Use the product property to simplify expressions with multiple logarithms.
  • Use the quotient property to simplify expressions with division.
  • Use the power property to simplify expressions with exponents.
  • Practice, practice, practice! The more you practice, the more comfortable you will become with logarithmic expressions.

Additional Resources

  • Khan Academy: Logarithms
  • Mathway: Logarithmic Expressions
  • Wolfram Alpha: Logarithmic Expressions

We hope that this article has been helpful in your understanding of logarithmic expressions. If you have any further questions or need additional help, please don't hesitate to ask.