Use The Properties Of Logarithms To Rewrite The Expression As A Single Logarithm. Show Your Algebraic Steps.$\[ 7 \log_2 X + 4 \log_2 Y - \log_2 3 \\]

Feb 28, 2025 by ADMIN 151 views

**Using Properties of Logarithms to Rewrite an Expression**

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. One of the key properties of logarithms is the ability to combine multiple logarithmic expressions into a single logarithmic expression. In this article, we will explore how to use the properties of logarithms to rewrite the given expression as a single logarithm.

The Given Expression

The given expression is:

${ 7 \log_2 x + 4 \log_2 y - \log_2 3 }$

Our goal is to rewrite this expression as a single logarithm using the properties of logarithms.

Property 1: Product Rule

The product rule 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:

${ \log_b (m \cdot n) = \log_b m + \log_b n }$

We can use this property to rewrite the given expression by combining the logarithmic terms.

Step 1: Combine the Logarithmic Terms

Using the product rule, we can rewrite the given expression as:

${ 7 \log_2 x + 4 \log_2 y - \log_2 3 = \log_2 (x^7) + \log_2 (y^4) - \log_2 3 }$

Step 2: Apply the Quotient Rule

The quotient rule states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

${ \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n }$

We can use this property to rewrite the expression further.

Step 3: Combine the Logarithmic Terms Using the Quotient Rule

Using the quotient rule, we can rewrite the expression as:

${ \log_2 (x^7) + \log_2 (y^4) - \log_2 3 = \log_2 \left( \frac{x^7 \cdot y^4}{3} \right) }$

Step 4: Simplify the Expression

We can simplify the expression by combining the exponents of x and y.

${ \log_2 \left( \frac{x^7 \cdot y^4}{3} \right) = \log_2 \left( \frac{x^7 \cdot y^4}{3} \right) }$

Conclusion

In this article, we used the properties of logarithms to rewrite the given expression as a single logarithm. We applied the product rule and the quotient rule to combine the logarithmic terms and simplify the expression. The final expression is:

${ \log_2 \left( \frac{x^7 \cdot y^4}{3} \right) }$

This expression represents the original expression as a single logarithm, using the properties of logarithms.

Key Takeaways

The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
The quotient rule states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.
We can use these properties to rewrite logarithmic expressions as single logarithms.

Practice Problems

Rewrite the expression 3 log_2 x + 2 log_2 y - log_2 4 as a single logarithm using the properties of logarithms.
Rewrite the expression 5 log_3 x - 2 log_3 y + log_3 9 as a single logarithm using the properties of logarithms.

References

[1] "Logarithms" by Khan Academy
[2] "Properties of Logarithms" by Math Open Reference

Introduction

Logarithms are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will answer some frequently asked questions about logarithm properties.

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:

${ \log_b (m \cdot n) = \log_b m + \log_b n }$

Q: What is the quotient rule of logarithms?

A: The quotient rule of logarithms states that the logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. Mathematically, this can be expressed as:

${ \log_b \left( \frac{m}{n} \right) = \log_b m - \log_b n }$

Q: How do I apply the product rule to rewrite a logarithmic expression?

A: To apply the product rule, you need to identify the product inside the logarithm and rewrite it as the sum of the logarithms of the individual factors. For example, if you have the expression log_2 (x * y), you can rewrite it as log_2 x + log_2 y.

Q: How do I apply the quotient rule to rewrite a logarithmic expression?

A: To apply the quotient rule, you need to identify the quotient inside the logarithm and rewrite it as the logarithm of the dividend minus the logarithm of the divisor. For example, if you have the expression log_2 (x / y), you can rewrite it as log_2 x - log_2 y.

Q: Can I use the product rule and quotient rule together to rewrite a logarithmic expression?

A: Yes, you can use the product rule and quotient rule together to rewrite a logarithmic expression. For example, if you have the expression log_2 (x * y / z), you can first apply the product rule to rewrite it as log_2 x + log_2 y - log_2 z.

Q: What is the power rule of logarithms?

A: The power rule of logarithms states that the logarithm of a power is equal to the exponent times the logarithm of the base. Mathematically, this can be expressed as:

${ \log_b (m^n) = n \log_b m }$

Q: How do I apply the power rule to rewrite a logarithmic expression?

A: To apply the power rule, you need to identify the power inside the logarithm and rewrite it as the exponent times the logarithm of the base. For example, if you have the expression log_2 (x^3), you can rewrite it as 3 log_2 x.

Q: Can I use the product rule, quotient rule, and power rule together to rewrite a logarithmic expression?

A: Yes, you can use the product rule, quotient rule, and power rule together to rewrite a logarithmic expression. For example, if you have the expression log_2 (x^3 * y^2 / z), you can first apply the power rule to rewrite it as 3 log_2 x + 2 log_2 y - log_2 z, and then apply the product rule and quotient rule to simplify the expression further.

Conclusion

In this article, we answered some frequently asked questions about logarithm properties. We discussed the product rule, quotient rule, and power rule, and provided examples of how to apply these rules to rewrite logarithmic expressions. By understanding these properties, you can simplify complex logarithmic expressions and solve mathematical problems with ease.

Practice Problems

Rewrite the expression log_2 (x * y / z) using the product rule and quotient rule.
Rewrite the expression log_2 (x^3 * y^2 / z) using the power rule, product rule, and quotient rule.
Rewrite the expression log_3 (x / y) using the quotient rule.

References

[1] "Logarithms" by Khan Academy
[2] "Properties of Logarithms" by Math Open Reference

Note: The references provided are for informational purposes only and are not directly related to the content of this article.