Which Of These Expressions Is Equivalent To $\log (20 \cdot 27$\]?A. $\log (20) - \log (27$\] B. $\log (20) \cdot \log (27$\] C. $20 \cdot \log (27$\] D. $\log (20) + \log (27$\]

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, particularly in algebra and calculus. They are used to represent the power to which a base number must be raised to produce a given value. In this article, we will explore the concept of logarithmic expressions and determine which of the given options is equivalent to log⁡(20⋅27)\log (20 \cdot 27).

What are Logarithmic Expressions?

A logarithmic expression is a mathematical operation that represents the inverse of exponentiation. It is a way of expressing a number in terms of its logarithm, which is the power to which a base number must be raised to produce that number. For example, if we have the expression log⁡2(8)\log_2 (8), it means that we are looking for the power to which 2 must be raised to produce 8. In this case, the answer is 3, because 23=82^3 = 8.

Properties of Logarithmic Expressions

There are several properties of logarithmic expressions that are essential to understand. One of the most important properties is the product rule, which states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. This can be expressed mathematically as:

log⁡(a⋅b)=log⁡(a)+log⁡(b)\log (a \cdot b) = \log (a) + \log (b)

This property is crucial in solving logarithmic expressions, as it allows us to break down complex expressions into simpler ones.

Applying the Product Rule

Now that we have understood the product rule, let's apply it to the given expression log⁡(20⋅27)\log (20 \cdot 27). Using the product rule, we can rewrite this expression as:

log⁡(20⋅27)=log⁡(20)+log⁡(27)\log (20 \cdot 27) = \log (20) + \log (27)

This is the correct answer, and it is option D.

Why are the Other Options Incorrect?

Let's examine the other options to understand why they are incorrect.

Option A: log⁡(20)−log⁡(27)\log (20) - \log (27)

This option is incorrect because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers, not the difference.

Option B: log⁡(20)⋅log⁡(27)\log (20) \cdot \log (27)

This option is incorrect because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers, not the product of the logarithms.

Option C: 20⋅log⁡(27)20 \cdot \log (27)

This option is incorrect because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers, not the product of one of the numbers and the logarithm of the other.

Conclusion

In conclusion, the correct answer is option D: log⁡(20)+log⁡(27)\log (20) + \log (27). This is because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. The other options are incorrect because they do not follow the product rule.

Final Thoughts

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is essential to solving complex mathematical problems. The product rule is a crucial property that allows us to break down complex expressions into simpler ones. By applying the product rule, we can determine which of the given options is equivalent to log⁡(20⋅27)\log (20 \cdot 27).

References

  • [1] "Logarithmic Expressions" by Math Open Reference
  • [2] "Properties of Logarithmic Expressions" by Khan Academy

Additional Resources

  • [1] "Logarithmic Expressions" by Wolfram MathWorld
  • [2] "Properties of Logarithmic Expressions" by MIT OpenCourseWare
    Logarithmic Expressions Q&A =============================

Frequently Asked Questions

Q: What is the product rule for logarithmic expressions?

A: The product rule for logarithmic expressions states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers. This can be expressed mathematically as:

log⁡(a⋅b)=log⁡(a)+log⁡(b)\log (a \cdot b) = \log (a) + \log (b)

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

A: To apply the product rule, simply break down the complex expression into simpler ones by using the product rule. For example, if we have the expression log⁡(20⋅27)\log (20 \cdot 27), we can rewrite it as:

log⁡(20⋅27)=log⁡(20)+log⁡(27)\log (20 \cdot 27) = \log (20) + \log (27)

Q: What is the difference between the product rule and the quotient rule for logarithmic expressions?

A: The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual numbers, while the quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual numbers. The quotient rule can be expressed mathematically as:

log⁡(ab)=log⁡(a)−log⁡(b)\log \left(\frac{a}{b}\right) = \log (a) - \log (b)

Q: How do I use the product rule to simplify a logarithmic expression?

A: To simplify a logarithmic expression using the product rule, simply break down the complex expression into simpler ones by using the product rule. For example, if we have the expression log⁡(20⋅27⋅3)\log (20 \cdot 27 \cdot 3), we can rewrite it as:

log⁡(20⋅27⋅3)=log⁡(20)+log⁡(27)+log⁡(3)\log (20 \cdot 27 \cdot 3) = \log (20) + \log (27) + \log (3)

Q: Can I use the product rule to simplify a logarithmic expression with a negative number?

A: Yes, you can use the product rule to simplify a logarithmic expression with a negative number. For example, if we have the expression log⁡(−20⋅27)\log (-20 \cdot 27), we can rewrite it as:

log⁡(−20⋅27)=log⁡(−20)+log⁡(27)\log (-20 \cdot 27) = \log (-20) + \log (27)

Q: How do I use the product rule to solve a logarithmic equation?

A: To solve a logarithmic equation using the product rule, simply apply the product rule to both sides of the equation. For example, if we have the equation log⁡(20⋅27)=log⁡(100)\log (20 \cdot 27) = \log (100), we can rewrite it as:

log⁡(20)+log⁡(27)=log⁡(100)\log (20) + \log (27) = \log (100)

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

A: Yes, you can use the product rule to solve a logarithmic inequality. For example, if we have the inequality log⁡(20⋅27)>log⁡(100)\log (20 \cdot 27) > \log (100), we can rewrite it as:

log⁡(20)+log⁡(27)>log⁡(100)\log (20) + \log (27) > \log (100)

Conclusion

In conclusion, the product rule for logarithmic expressions is a powerful tool that allows us to simplify complex expressions and solve logarithmic equations and inequalities. By understanding the product rule and how to apply it, you can become more confident and proficient in solving logarithmic problems.

Additional Resources

  • [1] "Logarithmic Expressions" by Math Open Reference
  • [2] "Properties of Logarithmic Expressions" by Khan Academy
  • [3] "Logarithmic Equations and Inequalities" by Wolfram MathWorld
  • [4] "Logarithmic Expressions" by MIT OpenCourseWare