Why Options A and C are Incorrect

Oct 13, 2025 by leolavanderia.com.br 138 views

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the properties of logarithms, specifically the expansion of $\log _h(9jk)$ . We will explore the correct expansion of this expression and discuss the implications of each option.

What are Logarithms?

A logarithm is the inverse operation of exponentiation. It is a mathematical function that takes a number as input and returns the exponent to which a base number must be raised to produce the input number. In other words, if $y = b^x$ , then $x = \log_b y$ . The logarithm of a number is a measure of the power to which the base must be raised to produce that number.

Properties of Logarithms

Logarithms have several important properties that make them useful in mathematics. Some of the key properties of logarithms include:

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

Expanding $\log _h(9jk)$

Now that we have discussed the properties of logarithms, let's apply them to the expression $\log _h(9jk)$ . We can use the product property to expand this expression as follows:

\log _h(9jk) = \log _h(9 \cdot j \cdot k) = \log _h 9 + \log _h j + \log _h k

This expansion is based on the product property of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Evaluating the Options

Now that we have expanded $\log _h(9jk)$ , let's evaluate the options provided:

Option A: $\log _h 9 \cdot \log _h j \cdot \log _h k$
Option B: $\log _h 9 + \log _h j + \log _h k$
Option C: $\log 9 + \log j + \log k$

Based on our expansion of $\log _h(9jk)$ , we can see that option B is the correct expansion of this expression.

Why Option B is Correct

Option B is the correct expansion of $\log _h(9jk)$ because it is based on the product property of logarithms. This property states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. By applying this property, we can expand $\log _h(9jk)$ as $\log _h 9 + \log _h j + \log _h k$ .

Why Options A and C are Incorrect

Options A and C are incorrect because they do not follow the product property of logarithms. Option A multiplies the logarithms of the individual factors, which is not correct. Option C uses the base 10 logarithm, which is not specified in the original expression.

Conclusion

In conclusion, the correct expansion of $\log _h(9jk)$ is $\log _h 9 + \log _h j + \log _h k$ . This expansion is based on the product property of logarithms, which states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. By understanding the properties of logarithms, we can solve mathematical problems with confidence.

Final Thoughts

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By applying the product property of logarithms, we can expand expressions like $\log _h(9jk)$ and solve mathematical problems with confidence. Remember to always follow the properties of logarithms to ensure accurate results.

References

"Logarithms" by Math Is Fun
"Properties of Logarithms" by Purplemath
"Logarithmic Expansion" by Khan Academy

Additional Resources

Logarithm Calculator: A calculator that can help you calculate logarithms.
Logarithm Table: A table that lists the logarithms of various numbers.
Logarithm Tutorial: A tutorial that provides a comprehensive introduction to logarithms.
Logarithmic Properties: A Q&A Guide =====================================

Introduction

In our previous article, we discussed the properties of logarithms and how to expand expressions like $\log _h(9jk)$ . In this article, we will provide a Q&A guide to help you better understand logarithmic properties and how to apply them in mathematical problems.

Q: What is the product property of logarithms?

A: The product property of logarithms states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. In other words, $\log_b (xy) = \log_b x + \log_b y$ .

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

A: To apply the product property of logarithms, simply break down the product into its individual factors and take the logarithm of each factor separately. Then, add the logarithms of the individual factors to get the final result.

Q: What is the quotient property of logarithms?

A: The quotient property of logarithms states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. In other words, $\log_b \left(\frac{x}{y}\right) = \log_b x - \log_b y$ .

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

A: To apply the quotient property of logarithms, simply take the logarithm of the dividend and the logarithm of the divisor separately. Then, subtract the logarithm of the divisor from the logarithm of the dividend to get the final result.

Q: What is the power property of logarithms?

A: The power property of logarithms states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words, $\log_b x^y = y \log_b x$ .

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

A: To apply the power property of logarithms, simply take the logarithm of the base and multiply it by the exponent. This will give you the final result.

Q: Can I use logarithmic properties to simplify expressions?

A: Yes, you can use logarithmic properties to simplify expressions. By applying the product, quotient, and power properties of logarithms, you can break down complex expressions into simpler ones.

Q: How do I choose the correct logarithmic property to apply?

A: To choose the correct logarithmic property to apply, simply look at the expression you are working with and identify the operation that is being performed. If the expression involves a product, use the product property. If the expression involves a quotient, use the quotient property. If the expression involves a power, use the power property.

Q: What are some common mistakes to avoid when working with logarithmic properties?

A: Some common mistakes to avoid when working with logarithmic properties include:

Forgetting to apply the product property: Make sure to break down products into their individual factors and take the logarithm of each factor separately.
Forgetting to apply the quotient property: Make sure to take the logarithm of the dividend and the logarithm of the divisor separately and subtract the logarithm of the divisor from the logarithm of the dividend.
Forgetting to apply the power property: Make sure to take the logarithm of the base and multiply it by the exponent.

Conclusion

In conclusion, logarithmic properties are a powerful tool for simplifying expressions and solving mathematical problems. By understanding the product, quotient, and power properties of logarithms, you can break down complex expressions into simpler ones and solve problems with confidence.

Final Thoughts

Logarithmic properties are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By applying the product, quotient, and power properties of logarithms, you can simplify expressions and solve problems with ease. Remember to always choose the correct logarithmic property to apply and avoid common mistakes.

References

"Logarithms" by Math Is Fun
"Properties of Logarithms" by Purplemath
"Logarithmic Expansion" by Khan Academy

Additional Resources

Logarithm Calculator: A calculator that can help you calculate logarithms.
Logarithm Table: A table that lists the logarithms of various numbers.
Logarithm Tutorial: A tutorial that provides a comprehensive introduction to logarithms.