Which Of These Expressions Is Equivalent To $\log (6 \cdot 7)$?A. $\log (6)-\log (7)$B. \$6 \cdot \log (7)$[/tex\]C. $\log (6) \cdot \log (7)$D. $\log (6)+\log (7)$

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In this article, we will explore the concept of logarithmic expressions and determine which of the given expressions is equivalent to $\log (6 \cdot 7)$.

What are Logarithmic Expressions?

A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to produce a given value. In other words, it is the inverse operation of exponentiation. The general form of a logarithmic expression is $\log_b (x)$, where $b$ is the base and $x$ is the value.

Properties of Logarithmic Expressions

There are several properties of logarithmic expressions that are essential to understand. These properties include:

  • Product Rule: $\log_b (x \cdot y) = \log_b (x) + \log_b (y)$
  • Quotient Rule: $\log_b (\frac{x}{y}) = \log_b (x) - \log_b (y)$
  • Power Rule: $\log_b (x^y) = y \cdot \log_b (x)$

Evaluating the Given Expression

Now that we have a good understanding of logarithmic expressions and their properties, let's evaluate the given expression $\log (6 \cdot 7)$. Using the product rule, we can rewrite this expression as:

log⁡(6⋅7)=log⁡(6)+log⁡(7)\log (6 \cdot 7) = \log (6) + \log (7)

Comparing the Options

Now that we have evaluated the given expression, let's compare it with the options provided:

  • Option A: $\log (6) - \log (7)$
  • Option B: $6 \cdot \log (7)$
  • Option C: $\log (6) \cdot \log (7)$
  • Option D: $\log (6) + \log (7)$

Conclusion

Based on our evaluation of the given expression and comparison with the options, we can conclude that the correct answer is:

  • Option D: $\log (6) + \log (7)$

This is because the product rule states that $\log_b (x \cdot y) = \log_b (x) + \log_b (y)$, which is exactly what we obtained when we evaluated the given expression.

Final Thoughts

In conclusion, understanding logarithmic expressions and their properties is crucial for solving various mathematical problems. By applying the product rule, we can rewrite the given expression $\log (6 \cdot 7)$ as $\log (6) + \log (7)$. This demonstrates the importance of logarithmic expressions in mathematics and highlights the need for a thorough understanding of their properties.

Common Mistakes to Avoid

When working with logarithmic expressions, it's essential to avoid common mistakes such as:

  • Misapplying the product rule: Make sure to apply the product rule correctly, as it can lead to incorrect results.
  • Forgetting the base: Always specify the base of the logarithmic expression to avoid confusion.
  • Not using the correct property: Use the correct property of logarithmic expressions to evaluate the given expression.

Real-World Applications

Logarithmic expressions have numerous real-world applications, including:

  • Finance: Logarithmic expressions are used to calculate interest rates and investment returns.
  • Science: Logarithmic expressions are used to model population growth and decay.
  • Engineering: Logarithmic expressions are used to design and optimize systems.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By applying the product rule, we can rewrite the given expression $\log (6 \cdot 7)$ as $\log (6) + \log (7)$. This demonstrates the importance of logarithmic expressions in mathematics and highlights the need for a thorough understanding of their properties.

Frequently Asked Questions

  • What is the product rule of logarithmic expressions?
    • The product rule states that $\log_b (x \cdot y) = \log_b (x) + \log_b (y)$.
  • How do I apply the product rule?
    • To apply the product rule, simply rewrite the given expression as the sum of two logarithmic expressions.
  • What are some common mistakes to avoid when working with logarithmic expressions?
    • Some common mistakes to avoid include misapplying the product rule, forgetting the base, and not using the correct property.

Additional Resources

For further learning, we recommend the following resources:

  • Textbooks: "Calculus" by Michael Spivak and "Mathematics for Computer Science" by Eric Lehman and Tom Leighton.
  • Online Courses: "Calculus" by MIT OpenCourseWare and "Mathematics for Computer Science" by Stanford University.
  • Websites: Khan Academy and Wolfram Alpha.
    Logarithmic Expressions Q&A =============================

Frequently Asked Questions

Q: What is the product rule of logarithmic expressions?

A: The product rule states that $\log_b (x \cdot y) = \log_b (x) + \log_b (y)$.

Q: How do I apply the product rule?

A: To apply the product rule, simply rewrite the given expression as the sum of two logarithmic expressions.

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

A: Some common mistakes to avoid include misapplying the product rule, forgetting the base, and not using the correct property.

Q: How do I evaluate a logarithmic expression with a variable base?

A: To evaluate a logarithmic expression with a variable base, you need to use the change of base formula: $\log_b (x) = \frac{\log_c (x)}{\log_c (b)}$, where $c$ is any positive real number not equal to 1.

Q: What is the difference between a logarithmic expression and an exponential expression?

A: A logarithmic expression is the inverse operation of an exponential expression. In other words, $\log_b (x)$ is the power to which the base $b$ must be raised to produce the value $x$, while $b^x$ is the value that results from raising the base $b$ to the power of $x$.

Q: Can I use the product rule to evaluate a logarithmic expression with a quotient?

A: No, the product rule only applies to logarithmic expressions with a product. To evaluate a logarithmic expression with a quotient, you need to use the quotient rule: $\log_b (\frac{x}{y}) = \log_b (x) - \log_b (y)$.

Q: How do I simplify a logarithmic expression with multiple terms?

A: To simplify a logarithmic expression with multiple terms, you can use the properties of logarithmic expressions, such as the product rule and the quotient rule, to combine the terms.

Q: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. However, make sure to check the calculator's settings and ensure that it is set to the correct base.

Q: What is the significance of logarithmic expressions in real-world applications?

A: Logarithmic expressions have numerous real-world applications, including finance, science, and engineering. They are used to model population growth and decay, calculate interest rates and investment returns, and design and optimize systems.

Additional Resources

For further learning, we recommend the following resources:

  • Textbooks: "Calculus" by Michael Spivak and "Mathematics for Computer Science" by Eric Lehman and Tom Leighton.
  • Online Courses: "Calculus" by MIT OpenCourseWare and "Mathematics for Computer Science" by Stanford University.
  • Websites: Khan Academy and Wolfram Alpha.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By applying the product rule and other properties of logarithmic expressions, we can rewrite and simplify complex expressions. We hope this Q&A article has provided you with a better understanding of logarithmic expressions and their applications.