What Is The First Step In Establishing That Log ⁡ ( 10 7 ⋅ 5 ) = 7 + Log ⁡ 5 \log \left(10^7 \cdot 5\right) = 7 + \log 5 Lo G ( 1 0 7 ⋅ 5 ) = 7 + Lo G 5 ?A. The First Step Is To Rewrite The Expression Using Multiplication, Such That Log ⁡ ( 10 7 ⋅ 5 ) = Log ⁡ 10 7 ⋅ Log ⁡ 5 \log \left(10^7 \cdot 5\right) = \log 10^7 \cdot \log 5 Lo G ( 1 0 7 ⋅ 5 ) = Lo G 1 0 7 ⋅ Lo G 5 .B. The

Mar 13, 2025 by ADMIN 380 views

**Understanding the Properties of Logarithms: A Step-by-Step Guide**

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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 first step in establishing the equation $\log \left(10^7 \cdot 5\right) = 7 + \log 5$ . We will delve into the properties of logarithms, specifically the product rule, and demonstrate how it can be applied to rewrite the given expression.

The Product Rule of Logarithms

The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as:

\log (ab) = \log a + \log b

This rule is a fundamental property of logarithms and is widely used in various mathematical applications.

Applying the Product Rule

To establish the equation $\log \left(10^7 \cdot 5\right) = 7 + \log 5$ , we need to apply the product rule of logarithms. The first step is to rewrite the expression using multiplication, such that:

\log \left(10^7 \cdot 5\right) = \log 10^7 \cdot \log 5

This is the correct application of the product rule, where we have broken down the product $10^7 \cdot 5$ into two separate factors, $10^7$ and $5$ , and taken the logarithm of each factor separately.

Why This Step is Essential

This step is essential because it allows us to apply the product rule of logarithms, which is a fundamental property of logarithms. By breaking down the product into two separate factors, we can then apply the product rule to rewrite the expression in a more manageable form.

Common Mistakes to Avoid

When applying the product rule, it's essential to remember that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This means that we need to break down the product into two separate factors and take the logarithm of each factor separately.

Conclusion

In conclusion, the first step in establishing the equation $\log \left(10^7 \cdot 5\right) = 7 + \log 5$ is to rewrite the expression using multiplication, such that $\log \left(10^7 \cdot 5\right) = \log 10^7 \cdot \log 5$ . This is the correct application of the product rule of logarithms, and it's essential for solving various mathematical problems.

Additional Tips and Tricks

When applying the product rule, make sure to break down the product into two separate factors.
Take the logarithm of each factor separately.
Use the product rule to rewrite the expression in a more manageable form.

Real-World Applications

The product rule of logarithms has numerous real-world applications, including:

Engineering: The product rule is used to calculate the logarithm of complex systems, such as electrical circuits and mechanical systems.
Computer Science: The product rule is used in algorithms for solving complex mathematical problems, such as linear algebra and optimization.
Finance: The product rule is used to calculate the logarithm of financial instruments, such as stocks and bonds.

Final Thoughts

In conclusion, the product rule of logarithms is a fundamental property of logarithms that is widely used in various mathematical applications. By understanding the product rule and applying it correctly, we can solve complex mathematical problems and make informed decisions in various fields.

References

[1] "Logarithms" by Math Is Fun
[2] "Product Rule of Logarithms" by Khan Academy
[3] "Logarithmic Properties" by Wolfram MathWorld

Glossary

Logarithm: A mathematical function that calculates the power to which a base number must be raised to produce a given value.
Product Rule: A property of logarithms that states the logarithm of a product is equal to the sum of the logarithms of the individual factors.
Base Number: A number that is used as the base for a logarithmic function.
Logarithmic Function: A mathematical function that calculates the logarithm of a given value.

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Q: What is the product rule of logarithms?

A: The product rule of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. Mathematically, this can be expressed as:

\log (ab) = \log a + \log b

Q: How do I apply the product rule of logarithms?

A: To apply the product rule, you need to break down the product into two separate factors and take the logarithm of each factor separately. For example, if you have the expression $\log (10^7 \cdot 5)$ , you can break it down into $\log 10^7 + \log 5$ .

Q: What is the difference between the product rule and the power rule of logarithms?

A: The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. The power rule, on the other hand, states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. Mathematically, this can be expressed as:

\log a^b = b \log a

Q: Can I use the product rule to rewrite an expression with multiple factors?

A: Yes, you can use the product rule to rewrite an expression with multiple factors. For example, if you have the expression $\log (10^7 \cdot 5 \cdot 2)$ , you can break it down into $\log 10^7 + \log 5 + \log 2$ .

Q: What are some common mistakes to avoid when applying the product rule?

A: Some common mistakes to avoid when applying the product rule include:

Not breaking down the product into separate factors
Not taking the logarithm of each factor separately
Using the product rule incorrectly, such as adding the exponents instead of the logarithms

Q: How do I use the product rule in real-world applications?

A: The product rule is used in various real-world applications, including:

Engineering: The product rule is used to calculate the logarithm of complex systems, such as electrical circuits and mechanical systems.
Computer Science: The product rule is used in algorithms for solving complex mathematical problems, such as linear algebra and optimization.
Finance: The product rule is used to calculate the logarithm of financial instruments, such as stocks and bonds.

Q: Can I use the product rule to solve logarithmic equations?

A: Yes, you can use the product rule to solve logarithmic equations. For example, if you have the equation $\log (10^7 \cdot 5) = 7 + \log 5$ , you can use the product rule to rewrite the expression as $\log 10^7 + \log 5 = 7 + \log 5$ .

Q: What are some additional tips and tricks for working with logarithms?

A: Some additional tips and tricks for working with logarithms include:

Using the product rule to rewrite expressions with multiple factors
Using the power rule to rewrite expressions with powers
Using the logarithmic identity $\log a^b = b \log a$ to simplify expressions

Q: How do I know when to use the product rule versus the power rule?

A: You should use the product rule when you have an expression with multiple factors, and you should use the power rule when you have an expression with a power. For example, if you have the expression $\log (10^7 \cdot 5)$ , you should use the product rule to rewrite it as $\log 10^7 + \log 5$ . If you have the expression $\log 10^{7 \cdot 5}$ , you should use the power rule to rewrite it as $7 \cdot 5 \log 10$ .

Q: Can I use the product rule to solve logarithmic inequalities?

A: Yes, you can use the product rule to solve logarithmic inequalities. For example, if you have the inequality $\log (10^7 \cdot 5) > 7 + \log 5$ , you can use the product rule to rewrite the expression as $\log 10^7 + \log 5 > 7 + \log 5$ .

Q: What are some common applications of logarithms in real-world scenarios?

A: Logarithms have numerous applications in real-world scenarios, including:

Engineering: Logarithms are used to calculate the logarithm of complex systems, such as electrical circuits and mechanical systems.
Computer Science: Logarithms are used in algorithms for solving complex mathematical problems, such as linear algebra and optimization.
Finance: Logarithms are used to calculate the logarithm of financial instruments, such as stocks and bonds.
Biology: Logarithms are used to calculate the logarithm of population growth and decay.
Medicine: Logarithms are used to calculate the logarithm of drug concentrations and dosages.

Q: Can I use logarithms to solve problems in other areas of mathematics?

A: Yes, you can use logarithms to solve problems in other areas of mathematics, including:

Algebra: Logarithms are used to solve equations and inequalities involving logarithmic functions.
Geometry: Logarithms are used to calculate the logarithm of geometric shapes, such as circles and spheres.
Trigonometry: Logarithms are used to calculate the logarithm of trigonometric functions, such as sine and cosine.

Q: What are some additional resources for learning about logarithms?

A: Some additional resources for learning about logarithms include:

Textbooks: There are many textbooks available that cover logarithms in detail.
Online resources: There are many online resources available that provide tutorials and examples on logarithms.
Video lectures: There are many video lectures available that provide an in-depth explanation of logarithms.
Practice problems: There are many practice problems available that allow you to practice working with logarithms.