Which Of These Expressions Is Equivalent To \log \left(\frac{12}{5}\right ]?A. Log ⁡ ( 12 ) + Log ⁡ ( 5 \log (12) + \log (5 Lo G ( 12 ) + Lo G ( 5 ] B. Log ⁡ ( 12 ) − Log ⁡ ( 5 \log (12) - \log (5 Lo G ( 12 ) − Lo G ( 5 ] C. Log ⁡ ( 12 ) ⋅ Log ⁡ ( 5 \log (12) \cdot \log (5 Lo G ( 12 ) ⋅ Lo G ( 5 ] D. 12 ⋅ Log ⁡ ( 5 12 \cdot \log (5 12 ⋅ Lo G ( 5 ]

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 $\log \left(\frac{12}{5}\right)$.

What are Logarithmic Expressions?

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

Properties of Logarithmic Expressions

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

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

Analyzing the Options

Now that we have a good understanding of logarithmic expressions and their properties, let's analyze the given options:

Option A: $\log (12) + \log (5)$

This option is attempting to apply the product rule, which states that $\log_b(xy) = \log_b(x) + \log_b(y)$. However, in this case, we are dealing with a quotient, not a product. Therefore, this option is not equivalent to $\log \left(\frac{12}{5}\right)$.

Option B: $\log (12) - \log (5)$

This option is attempting to apply the quotient rule, which states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. This is the correct property to apply in this case, and it is indeed equivalent to $\log \left(\frac{12}{5}\right)$.

Option C: $\log (12) \cdot \log (5)$

This option is attempting to apply the power rule, which states that $\log_b(x^y) = y \cdot \log_b(x)$. However, in this case, we are dealing with a quotient, not a power. Therefore, this option is not equivalent to $\log \left(\frac{12}{5}\right)$.

Option D: $12 \cdot \log (5)$

This option is attempting to apply the power rule, but it is incorrectly applying the property. The correct application of the power rule would be $\log (12^5)$, not $12 \cdot \log (5)$. Therefore, this option is not equivalent to $\log \left(\frac{12}{5}\right)$.

Conclusion

In conclusion, the correct answer is option B: $\log (12) - \log (5)$. This option correctly applies the quotient rule, which states that $\log_b(\frac{x}{y}) = \log_b(x) - \log_b(y)$. This is indeed equivalent to $\log \left(\frac{12}{5}\right)$.

Final Thoughts

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is essential for solving problems in algebra and calculus. By applying the correct properties, we can determine which of the given options is equivalent to $\log \left(\frac{12}{5}\right)$. In this article, we have explored the concept of logarithmic expressions and determined that option B is the correct answer.

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: A Q&A Guide =====================================

Introduction

Logarithmic expressions are a fundamental concept in mathematics, particularly in algebra and calculus. In our previous article, we explored the concept of logarithmic expressions and determined which of the given options is equivalent to $\log \left(\frac{12}{5}\right)$. In this article, we will provide a Q&A guide to help you better understand logarithmic expressions and their properties.

Q: What is a logarithmic expression?

A: A logarithmic expression is a mathematical operation that represents the inverse of exponentiation. In other words, it is the power to which a base number must be raised to produce a given value.

Q: What are the properties of logarithmic expressions?

A: There are several properties of logarithmic expressions, including:

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

Q: How do I apply the product rule?

A: To apply the product rule, you need to multiply the logarithms of the two numbers. For example, if you have $\log_b(xy)$, you would calculate $\log_b(x) + \log_b(y)$.

Q: How do I apply the quotient rule?

A: To apply the quotient rule, you need to subtract the logarithm of the divisor from the logarithm of the dividend. For example, if you have $\log_b(\frac{x}{y})$, you would calculate $\log_b(x) - \log_b(y)$.

Q: How do I apply the power rule?

A: To apply the power rule, you need to multiply the exponent by the logarithm of the base. For example, if you have $\log_b(x^y)$, you would calculate $y \cdot \log_b(x)$.

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

A: A logarithmic expression represents the power to which a base number must be raised to produce a given value, while an exponential expression represents the result of raising a base number to a given power.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you need to apply the properties of logarithmic expressions, such as the product rule, quotient rule, and power rule.

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

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

• Confusing the product rule with the quotient rule
• Failing to apply the correct property
• Not simplifying the expression

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is essential for solving problems in algebra and calculus. By applying the correct properties and avoiding common mistakes, you can simplify logarithmic expressions and solve problems with confidence.

Final Thoughts

Logarithmic expressions are a powerful tool in mathematics, and understanding their properties is essential for success in algebra and calculus. By practicing and applying the properties of logarithmic expressions, you can become proficient in simplifying and solving logarithmic expressions.

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