Applying the Power Property

Oct 13, 2025 by windailys.com 52 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 focus on rewriting logarithmic expressions using specific properties. We will examine the given expression, $\log \sqrt[20]{625 x^4} = \frac{1}{5} \log 5 x$ , and determine the property or properties used to rewrite it.

The Given Expression

The given expression is $\log \sqrt[20]{625 x^4} = \frac{1}{5} \log 5 x$ . To rewrite this expression, we need to apply the properties of logarithms.

Properties of Logarithms

There are four main properties of logarithms:

Product Property: $\log (ab) = \log a + \log b$
Power Property: $\log a^b = b \log a$
Commutative Property: $\log a = \log b$ if and only if $a = b$
Quotient Property: $\log \frac{a}{b} = \log a - \log b$

Rewriting the Expression

To rewrite the given expression, we can start by applying the Power Property. The expression $\sqrt[20]{625 x^4}$ can be rewritten as $(625 x^4)^{\frac{1}{20}}$ . Using the Power Property, we can rewrite this expression as $\frac{1}{20} \log (625 x^4)$ .

Applying the Power Property

Now, we can apply the Power Property to the expression $\frac{1}{20} \log (625 x^4)$ . Using the Power Property, we can rewrite this expression as $\frac{1}{20} (\log 625 + \log x^4)$ .

Simplifying the Expression

We can simplify the expression $\frac{1}{20} (\log 625 + \log x^4)$ by applying the Product Property. The expression $\log 625 + \log x^4$ can be rewritten as $\log (625 x^4)$ .

Rewriting the Expression Using the Product Property

Now, we can rewrite the expression $\frac{1}{20} (\log 625 + \log x^4)$ using the Product Property. The expression $\log (625 x^4)$ can be rewritten as $\log 625 + \log x^4$ .

Simplifying the Expression Further

We can simplify the expression $\log 625 + \log x^4$ by applying the Power Property. The expression $\log x^4$ can be rewritten as $4 \log x$ .

Rewriting the Expression Using the Power Property

Now, we can rewrite the expression $\log 625 + 4 \log x$ using the Power Property. The expression $\log 625$ can be rewritten as $\log 5^4$ .

Simplifying the Expression Further

We can simplify the expression $\log 5^4 + 4 \log x$ by applying the Power Property. The expression $\log 5^4$ can be rewritten as $4 \log 5$ .

Rewriting the Expression Using the Power Property

Now, we can rewrite the expression $4 \log 5 + 4 \log x$ using the Power Property. The expression $4 \log 5 + 4 \log x$ can be rewritten as $4 (\log 5 + \log x)$ .

Simplifying the Expression Further

We can simplify the expression $4 (\log 5 + \log x)$ by applying the Product Property. The expression $\log 5 + \log x$ can be rewritten as $\log (5 x)$ .

Rewriting the Expression Using the Product Property

Now, we can rewrite the expression $4 (\log 5 + \log x)$ using the Product Property. The expression $4 (\log 5 + \log x)$ can be rewritten as $4 \log (5 x)$ .

Conclusion

In conclusion, the given expression $\log \sqrt[20]{625 x^4} = \frac{1}{5} \log 5 x$ can be rewritten using the Power Property and the Product Property. The expression can be rewritten as $4 \log (5 x)$ .

Answer

The correct answer is B. Power Property.

Final Answer

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

References

[1] "Logarithms." Wikipedia, Wikimedia Foundation, 2023, en.wikipedia.org/wiki/Logarithm.
[2] "Properties of Logarithms." Math Is Fun, mathisfun.com/algebra/logarithms.html.
[3] "Logarithmic Identities." Purplemath, purplemath.com/modules/logid.htm.
Q&A: Properties of Logarithms =============================

Introduction

Q1: What is the Product Property of Logarithms?

A1: The Product Property of Logarithms states that $\log (ab) = \log a + \log b$ . This means that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Q2: What is the Power Property of Logarithms?

A2: The Power Property of Logarithms states that $\log a^b = b \log a$ . This means that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q3: What is the Commutative Property of Logarithms?

A3: The Commutative Property of Logarithms states that $\log a = \log b$ if and only if $a = b$ . This means that the logarithm of two numbers are equal if and only if the numbers themselves are equal.

Q4: What is the Quotient Property of Logarithms?

A4: The Quotient Property of Logarithms states that $\log \frac{a}{b} = \log a - \log b$ . This means that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

Q5: How do I apply the Product Property of Logarithms?

A5: To apply the Product Property of Logarithms, you need to follow these steps:

Identify the product inside the logarithm.
Break down the product into individual factors.
Take the logarithm of each factor separately.
Add the logarithms of the individual factors.

Q6: How do I apply the Power Property of Logarithms?

A6: To apply the Power Property of Logarithms, you need to follow these steps:

Identify the power inside the logarithm.
Break down the power into its base and exponent.
Take the logarithm of the base.
Multiply the logarithm of the base by the exponent.

Q7: How do I apply the Commutative Property of Logarithms?

A7: To apply the Commutative Property of Logarithms, you need to follow these steps:

Identify the two numbers inside the logarithm.
Check if the numbers are equal.
If the numbers are equal, then the logarithms are also equal.

Q8: How do I apply the Quotient Property of Logarithms?

A8: To apply the Quotient Property of Logarithms, you need to follow these steps:

Identify the quotient inside the logarithm.
Break down the quotient into its dividend and divisor.
Take the logarithm of the dividend.
Take the logarithm of the divisor.
Subtract the logarithm of the divisor from the logarithm of the dividend.

Conclusion

In conclusion, the properties of logarithms are essential for solving various mathematical problems. By understanding the Product Property, Power Property, Commutative Property, and Quotient Property, you can apply them to rewrite logarithmic expressions and solve problems.

References

[1] "Logarithms." Wikipedia, Wikimedia Foundation, 2023, en.wikipedia.org/wiki/Logarithm.
[2] "Properties of Logarithms." Math Is Fun, mathisfun.com/algebra/logarithms.html.
[3] "Logarithmic Identities." Purplemath, purplemath.com/modules/logid.htm.

Frequently Asked Questions

Q: What is the difference between the Product Property and the Power Property? A: The Product Property states that $\log (ab) = \log a + \log b$ , while the Power Property states that $\log a^b = b \log a$ .
Q: How do I apply the Commutative Property of Logarithms? A: To apply the Commutative Property of Logarithms, you need to check if the numbers inside the logarithm are equal.
Q: What is the Quotient Property of Logarithms? A: The Quotient Property of Logarithms states that $\log \frac{a}{b} = \log a - \log b$ .

Additional Resources

[1] "Logarithms." Khan Academy, khanacademy.org/math/algebra/x2f-logarithms.
[2] "Properties of Logarithms." IXL, ixl.com/math/logarithms.
[3] "Logarithmic Identities." Mathway, mathway.com/subject/logarithmic-identities.