Apply the quotient property

Oct 13, 2025 by wbrum.org 62 views

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on using properties of logarithms to evaluate the expression $\log_3 162 - \log_3 18$ . We will break down the problem step by step, explaining each concept and providing examples to illustrate the process.

Understanding Logarithms

Before we dive into the problem, let's briefly review the concept of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$ , then $\log_a x = y$ . The logarithm of a number $x$ with base $a$ is the exponent to which $a$ must be raised to produce $x$ .

Properties of Logarithms

There are several properties of logarithms that we will use to evaluate the expression $\log_3 162 - \log_3 18$ . These properties include:

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

Evaluating the Expression

Now that we have reviewed the properties of logarithms, let's evaluate the expression $\log_3 162 - \log_3 18$ .

Step 1: Factorize the Numbers

To evaluate the expression, we need to factorize the numbers $162$ and $18$ . We can write $162$ as $2 \cdot 81$ and $18$ as $2 \cdot 9$ .

import math
num1 = 162
num2 = 18
factor1 = 2
factor2 = 81
factor3 = 2
factor4 = 9

Step 2: Apply the Quotient Property

Using the quotient property, we can rewrite the expression as $\log_3 \left(\frac{2 \cdot 81}{2 \cdot 9}\right) = \log_3 \left(\frac{81}{9}\right)$ .

# Apply the quotient property
result = math.log(81/9, 3)

Step 3: Simplify the Expression

We can simplify the expression further by canceling out the common factors. In this case, we can cancel out the factor of $9$ in the numerator and denominator.

# Simplify the expression
result = math.log(9, 3)

Step 4: Evaluate the Logarithm

Finally, we can evaluate the logarithm using the definition of logarithms. Since $3^2 = 9$ , we have $\log_3 9 = 2$ .

# Evaluate the logarithm
result = 2

Conclusion

In this article, we used the properties of logarithms to evaluate the expression $\log_3 162 - \log_3 18$ . We broke down the problem step by step, explaining each concept and providing examples to illustrate the process. By applying the quotient property and simplifying the expression, we were able to evaluate the logarithm and arrive at the final answer.

Example Problems

Here are a few example problems that you can try to practice your skills:

Evaluate the expression $\log_2 16 - \log_2 4$
Evaluate the expression $\log_5 25 - \log_5 5$
Evaluate the expression $\log_7 49 - \log_7 7$

Solutions

Here are the solutions to the example problems:

$\log_2 16 - \log_2 4 = \log_2 \left(\frac{16}{4}\right) = \log_2 4 = 2$
$\log_5 25 - \log_5 5 = \log_5 \left(\frac{25}{5}\right) = \log_5 5 = 1$
$\log_7 49 - \log_7 7 = \log_7 \left(\frac{49}{7}\right) = \log_7 7 = 1$

Practice Problems

Here are a few practice problems that you can try to practice your skills:

Evaluate the expression $\log_3 27 - \log_3 9$
Evaluate the expression $\log_4 16 - \log_4 4$
Evaluate the expression $\log_6 36 - \log_6 6$

Solutions

Here are the solutions to the practice problems:

$\log_3 27 - \log_3 9 = \log_3 \left(\frac{27}{9}\right) = \log_3 3 = 1$
$\log_4 16 - \log_4 4 = \log_4 \left(\frac{16}{4}\right) = \log_4 4 = 2$
$\log_6 36 - \log_6 6 = \log_6 \left(\frac{36}{6}\right) = \log_6 6 = 1$

Conclusion

Introduction

Q: What is a logarithm?

A: A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$ , then $\log_a x = y$ . The logarithm of a number $x$ with base $a$ is the exponent to which $a$ must be raised to produce $x$ .

Q: What are the properties of logarithms?

A: There are several properties of logarithms that we will use to evaluate logarithmic expressions. These properties include:

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

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you need to follow these steps:

Factorize the numbers: Factorize the numbers in the expression to simplify it.
Apply the quotient property: Use the quotient property to rewrite the expression as a difference of logarithms.
Simplify the expression: Simplify the expression by canceling out common factors.
Evaluate the logarithm: Evaluate the logarithm using the definition of logarithms.

Q: What is the difference between a logarithm and an exponential function?

A: A logarithm is the inverse operation of an exponential function. In other words, if $x = a^y$ , then $\log_a x = y$ . An exponential function is a function of the form $f(x) = a^x$ , where $a$ is a positive constant.

Q: How do I use logarithmic properties to simplify an expression?

A: To use logarithmic properties to simplify an expression, you need to follow these steps:

Identify the properties: Identify the properties of logarithms that can be applied to the expression.
Apply the properties: Apply the properties to simplify the expression.
Simplify the expression: Simplify the expression by canceling out common factors.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

$\log_a x$
$\log_a (xy)$
$\log_a \left(\frac{x}{y}\right)$
$\log_a x^y$

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 follow these steps:

Rewrite the expression: Rewrite the expression with a positive exponent.
Apply the power property: Apply the power property to simplify the expression.
Evaluate the logarithm: Evaluate the logarithm using the definition of logarithms.

Q: What is the relationship between logarithms and exponents?

A: The relationship between logarithms and exponents is that they are inverse operations. In other words, if $x = a^y$ , then $\log_a x = y$ . This means that logarithms and exponents are related in a way that they can be used to solve equations and inequalities.

Conclusion

In this article, we provided a Q&A guide to help you understand logarithmic expressions and how to evaluate them. We covered topics such as the definition of logarithms, properties of logarithms, and how to evaluate logarithmic expressions. We also provided examples and practice problems to help you practice your skills.