difference of the logarithms

Oct 13, 2025 by magicalceylontours.com 38 views

Name: _______________________________________ PBL - Epidemics Explained

#8 - Discovering Properties of Logarithms

1. Match the expressions in Column A with the expressions in Column B

Column A	Column B
$\log_{a} (x)$	$\log_{a} (x) = y$
$\log_{a} (x) = y$	$\log_{a} (x)$
$\log_{a} (x) + \log_{a} (y)$	$\log_{a} (x) \cdot \log_{a} (y)$
$\log_{a} (x) \cdot \log_{a} (y)$	$\log_{a} (x) + \log_{a} (y)$
$\log_{a} (x) - \log_{a} (y)$	$\log_{a} (x) - \log_{a} (y)$

Instructions: Match the expressions in Column A with the expressions in Column B.

2. Properties of Logarithms

Logarithms are the inverse operation of exponentiation. They are used to solve equations that involve exponential functions. In this section, we will explore the properties of logarithms, which will help us to simplify and manipulate logarithmic expressions.

Property 1: The Logarithm of a Product

The logarithm of a product is equal to the sum of the logarithms of the individual factors. This property can be expressed as:

$\log_{a} (x \cdot y) = \log_{a} (x) + \log_{a} (y)$

Example: Find the value of $\log_{2} (8 \cdot 16)$ .

Using the property above, we can rewrite the expression as:

$\log_{2} (8 \cdot 16) = \log_{2} (8) + \log_{2} (16)$

We know that $\log_{2} (8) = 3$ and $\log_{2} (16) = 4$ , so:

$\log_{2} (8 \cdot 16) = 3 + 4 = 7$

Property 2: The Logarithm of a Quotient

The logarithm of a quotient is equal to the difference of the logarithms of the individual factors. This property can be expressed as:

$\log_{a} (\frac{x}{y}) = \log_{a} (x) - \log_{a} (y)$

Example: Find the value of $\log_{2} (\frac{8}{16})$ .

Using the property above, we can rewrite the expression as:

$\log_{2} (\frac{8}{16}) = \log_{2} (8) - \log_{2} (16)$

We know that $\log_{2} (8) = 3$ and $\log_{2} (16) = 4$ , so:

$\log_{2} (\frac{8}{16}) = 3 - 4 = -1$

Property 3: The Logarithm of a Power

The logarithm of a power is equal to the product of the logarithm of the base and the exponent. This property can be expressed as:

$\log_{a} (x^{n}) = n \cdot \log_{a} (x)$

Example: Find the value of $\log_{2} (8^{3})$ .

Using the property above, we can rewrite the expression as:

$\log_{2} (8^{3}) = 3 \cdot \log_{2} (8)$

We know that $\log_{2} (8) = 3$ , so:

$\log_{2} (8^{3}) = 3 \cdot 3 = 9$

3. Applications of Logarithms

Logarithms have many applications in various fields, including:

Mathematics: Logarithms are used to solve equations that involve exponential functions.
Science: Logarithms are used to measure the intensity of earthquakes and the magnitude of stars.
Engineering: Logarithms are used to design electronic circuits and to calculate the gain of amplifiers.
Finance: Logarithms are used to calculate the return on investment and to determine the value of a portfolio.

4. Conclusion

In this section, we have explored the properties of logarithms, which are used to simplify and manipulate logarithmic expressions. We have also seen some of the applications of logarithms in various fields. Logarithms are a powerful tool that can be used to solve equations that involve exponential functions, and they have many practical applications in science, engineering, and finance.

5. Exercises

Find the value of $\log_{2} (8 \cdot 16)$ .
Find the value of $\log_{2} (\frac{8}{16})$ .
Find the value of $\log_{2} (8^{3})$ .
Find the value of $\log_{2} (\frac{8^{3}}{16})$ .
Find the value of $\log_{2} (8 \cdot 16^{2})$ .

6. Answers

$\log_{2} (8 \cdot 16) = 7$
$\log_{2} (\frac{8}{16}) = -1$
$\log_{2} (8^{3}) = 9$
$\log_{2} (\frac{8^{3}}{16}) = 6$
$\log_{2} (8 \cdot 16^{2}) = 10$
Unit 2 - Exponentials & Logarithms =====================================

Name: _______________________________________ PBL - Epidemics Explained

#8 - Discovering Properties of Logarithms

7. Q&A

Q1: What is the inverse operation of exponentiation?

A1: The inverse operation of exponentiation is logarithm.

Q2: What is the property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual factors?

A2: The property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual factors is:

$\log_{a} (x \cdot y) = \log_{a} (x) + \log_{a} (y)$

Q3: What is the property of logarithms that states the logarithm of a quotient is equal to the difference of the logarithms of the individual factors?

A3: The property of logarithms that states the logarithm of a quotient is equal to the difference of the logarithms of the individual factors is:

$\log_{a} (\frac{x}{y}) = \log_{a} (x) - \log_{a} (y)$

Q4: What is the property of logarithms that states the logarithm of a power is equal to the product of the logarithm of the base and the exponent?

A4: The property of logarithms that states the logarithm of a power is equal to the product of the logarithm of the base and the exponent is:

$\log_{a} (x^{n}) = n \cdot \log_{a} (x)$

Q5: What are some of the applications of logarithms?

A5: Some of the applications of logarithms include:

Mathematics: Logarithms are used to solve equations that involve exponential functions.
Science: Logarithms are used to measure the intensity of earthquakes and the magnitude of stars.
Engineering: Logarithms are used to design electronic circuits and to calculate the gain of amplifiers.
Finance: Logarithms are used to calculate the return on investment and to determine the value of a portfolio.

8. Common Mistakes to Avoid

Mistake 1: Not using the correct property of logarithms.

Solution: Make sure to use the correct property of logarithms, such as the product rule, quotient rule, or power rule.

Mistake 2: Not simplifying the expression.

Solution: Simplify the expression by combining like terms and using the properties of logarithms.

Mistake 3: Not checking the domain of the logarithm.

Solution: Check the domain of the logarithm to make sure that the argument is positive.

9. Conclusion

In this section, we have explored the properties of logarithms and their applications. We have also seen some common mistakes to avoid when working with logarithms. Logarithms are a powerful tool that can be used to solve equations that involve exponential functions, and they have many practical applications in science, engineering, and finance.

10. Exercises

Find the value of $\log_{2} (8 \cdot 16)$ .
Find the value of $\log_{2} (\frac{8}{16})$ .
Find the value of $\log_{2} (8^{3})$ .
Find the value of $\log_{2} (\frac{8^{3}}{16})$ .
Find the value of $\log_{2} (8 \cdot 16^{2})$ .

11. Answers

$\log_{2} (8 \cdot 16) = 7$
$\log_{2} (\frac{8}{16}) = -1$
$\log_{2} (8^{3}) = 9$
$\log_{2} (\frac{8^{3}}{16}) = 6$
$\log_{2} (8 \cdot 16^{2}) = 10$