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)\]

Feb 25, 2025 by ADMIN 161 views

**Logarithmic Expressions: Simplifying and Equivalency**

Introduction

Logarithmic expressions are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to represent the power to which a base number must be raised to produce a given value. In this article, we will explore the concept of logarithmic expressions and focus on simplifying and finding equivalency between different expressions. We will examine a specific problem involving logarithmic expressions and determine which expression is equivalent to a given expression.

Understanding Logarithmic Expressions

A logarithmic expression is written in the form $\log_b a = c$ , where $b$ is the base, $a$ is the argument, and $c$ is the result. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, $\log_2 8 = 3$ because $2^3 = 8$ .

Properties of Logarithmic Expressions

There are several properties of logarithmic expressions that are essential to understand when simplifying and finding equivalency between expressions. These properties include:

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

Simplifying Logarithmic Expressions

To simplify a logarithmic expression, we can use the properties mentioned above. For example, consider the expression $\log 18 - \log (p+2)$ . We can simplify this expression using the quotient property.

Step 1: Identify the Quotient Property

The quotient property states that $\log_b \frac{x}{y} = \log_b x - \log_b y$ . In this case, we have $\log 18 - \log (p+2)$ , which can be rewritten as $\log \frac{18}{p+2}$ .

Step 2: Apply the Quotient Property

Using the quotient property, we can rewrite the expression as $\log \frac{18}{p+2}$ . This is the simplified form of the original expression.

Which Expression is Equivalent?

Now that we have simplified the expression, we can compare it to the given options to determine which one is equivalent.

Option A: $\log \frac{p+2}{18}$
Option B: $\log \frac{18}{p+2}$
Option C: $\log \frac{20}{p}$
Option D: $\log [18 \cdot (p+2)]$

Comparing Options

Let's compare the simplified expression to each of the options.

Option A: $\log \frac{p+2}{18}$ is not equivalent to the simplified expression because the order of the numerator and denominator is reversed.
Option B: $\log \frac{18}{p+2}$ is equivalent to the simplified expression because it has the same numerator and denominator.
Option C: $\log \frac{20}{p}$ is not equivalent to the simplified expression because the numerator and denominator are different.
Option D: $\log [18 \cdot (p+2)]$ is not equivalent to the simplified expression because it is not in the form of a quotient.

Conclusion

In conclusion, the expression $\log 18 - \log (p+2)$ is equivalent to $\log \frac{18}{p+2}$ . This is because we can use the quotient property to simplify the expression and rewrite it in the form of a quotient. The other options are not equivalent to the simplified expression.

Final Answer

The final answer is $\boxed{B}$ .

Additional Examples

Here are some additional examples of simplifying and finding equivalency between logarithmic expressions.

$\log 24 - \log (p+1) = \log \frac{24}{p+1}$
$\log 15 + \log (p+3) = \log [15 \cdot (p+3)]$
$\log \frac{20}{p} - \log 2 = \log \frac{10}{p}$

These examples demonstrate how to use the properties of logarithmic expressions to simplify and find equivalency between different expressions.

Conclusion

Introduction

In our previous article, we explored the concept of logarithmic expressions and how to simplify and find equivalency between them. In this article, we will answer some frequently asked questions about logarithmic expressions.

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

A: A logarithmic expression is written in the form $\log_b a = c$ , where $b$ is the base, $a$ is the argument, and $c$ is the result. An exponential expression is written in the form $b^c = a$ , where $b$ is the base, $c$ is the exponent, and $a$ is the result.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms, such as the product property, quotient property, and power property. For example, consider the expression $\log 18 - \log (p+2)$ . We can simplify this expression using the quotient property.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that $\log_b \frac{x}{y} = \log_b x - \log_b y$ . This means that we can rewrite a quotient of two logarithmic expressions as the difference of the two logarithmic expressions.

Q: How do I use the quotient property to simplify a logarithmic expression?

A: To use the quotient property to simplify a logarithmic expression, we can rewrite the expression as a quotient of two logarithmic expressions. For example, consider the expression $\log 18 - \log (p+2)$ . We can rewrite this expression as $\log \frac{18}{p+2}$ .

Q: What is the product property of logarithms?

A: The product property of logarithms states that $\log_b (xy) = \log_b x + \log_b y$ . This means that we can rewrite a product of two logarithmic expressions as the sum of the two logarithmic expressions.

Q: How do I use the product property to simplify a logarithmic expression?

A: To use the product property to simplify a logarithmic expression, we can rewrite the expression as a product of two logarithmic expressions. For example, consider the expression $\log 24 + \log (p+1)$ . We can rewrite this expression as $\log [24 \cdot (p+1)]$ .

Q: What is the power property of logarithms?

A: The power property of logarithms states that $\log_b x^y = y \log_b x$ . This means that we can rewrite a logarithmic expression with an exponent as the exponent times the logarithmic expression.

Q: How do I use the power property to simplify a logarithmic expression?

A: To use the power property to simplify a logarithmic expression, we can rewrite the expression as the exponent times the logarithmic expression. For example, consider the expression $\log 15^2$ . We can rewrite this expression as $2 \log 15$ .

Q: Can I use logarithmic expressions to solve equations?

A: Yes, logarithmic expressions can be used to solve equations. For example, consider the equation $\log 18 - \log (p+2) = 0$ . We can solve this equation by simplifying the expression and then solving for $p$ .

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, we can use the properties of logarithms to simplify the expression and then solve for the variable. For example, consider the equation $\log 18 - \log (p+2) = 0$ . We can solve this equation by simplifying the expression and then solving for $p$ .

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding their properties and how to simplify and find equivalency between them is essential. By using the quotient property, product property, and power property, we can simplify and rewrite logarithmic expressions in different forms. We can also use logarithmic expressions to solve equations.