Which Of These Expressions Is Equivalent To Log ⁡ ( 20 ⋅ 27 \log (20 \cdot 27 Lo G ( 20 ⋅ 27 ]?A. Log ⁡ ( 20 ) + Log ⁡ ( 27 \log (20) + \log (27 Lo G ( 20 ) + Lo G ( 27 ] B. Log ⁡ ( 20 ) − Log ⁡ ( 27 \log (20) - \log (27 Lo G ( 20 ) − Lo G ( 27 ] C. 20 ⋅ Log ⁡ ( 27 20 \cdot \log (27 20 ⋅ Lo G ( 27 ] D. Log ⁡ ( 20 ) ⋅ Log ⁡ ( 27 \log (20) \cdot \log (27 Lo G ( 20 ) ⋅ Lo G ( 27 ]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. In this article, we will explore the concept of logarithmic expressions and discuss which of the given expressions is equivalent to $\log (20 \cdot 27)$.

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 logarithmic function is denoted by the symbol $\log$ and is defined as:

$$\log_b(x) = y \iff b^y = x$$

where $b$ is the base of the logarithm, $x$ is the value, and $y$ is the logarithm of $x$ with base $b$.

Properties of Logarithmic Expressions

Logarithmic expressions have several properties that make them useful in various mathematical and real-world applications. Some of the key properties of logarithmic expressions include:

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

Which Expression is Equivalent to $\log (20 \cdot 27)$?

Now that we have discussed the properties of logarithmic expressions, let's examine the given options and determine which one is equivalent to $\log (20 \cdot 27)$.

Option A: $\log (20) + \log (27)$

Using the product rule, we can rewrite $\log (20 \cdot 27)$ as $\log (20) + \log (27)$. This is because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Option B: $\log (20) - \log (27)$

Using the quotient rule, we can rewrite $\log (20 \cdot 27)$ as $\log (20) - \log (27)$. However, this is not the correct expression, as the quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Option C: $20 \cdot \log (27)$

Using the power rule, we can rewrite $\log (20 \cdot 27)$ as $20 \cdot \log (27)$. However, this is not the correct expression, as the power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Option D: $\log (20) \cdot \log (27)$

This expression is not equivalent to $\log (20 \cdot 27)$, as it involves the product of two logarithms, which is not a valid operation.

Conclusion

In conclusion, the correct expression that is equivalent to $\log (20 \cdot 27)$ is $\log (20) + \log (27)$. This is because the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.

Final Answer

The final answer is: $\boxed{A}$

Additional Resources

For further reading on logarithmic expressions, we recommend the following resources:

• Wikipedia: Logarithm: A comprehensive article on logarithmic expressions, including their properties and applications.
• Khan Academy: Logarithms: A video tutorial on logarithmic expressions, including their properties and applications.
• Math Is Fun: Logarithms: A website that provides interactive lessons and exercises on logarithmic expressions.

FAQs

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: What is the product 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 factors.

Q: What is the quotient rule for logarithmic expressions? A: The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q&A: Frequently Asked Questions

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: What is the product 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 factors. In other words, $\log_b(x \cdot y) = \log_b(x) + \log_b(y)$.

Q: What is the quotient rule for logarithmic expressions? A: The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors. In other words, $\log_b(x \div y) = \log_b(x) - \log_b(y)$.

Q: What is the power rule for logarithmic expressions? A: The power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base. In other words, $\log_b(x^y) = y \cdot \log_b(x)$.

Q: How do I evaluate a logarithmic expression? A: To evaluate a logarithmic expression, you need to find the value of the logarithm. This can be done using a calculator or by using the properties of logarithms.

Q: What is the base of a logarithmic expression? A: The base of a logarithmic expression is the number that is raised to a power to produce the given value. For example, in the expression $\log_2(x)$, the base is 2.

Q: What is the logarithm of a number? A: The logarithm of a number is the power to which the base must be raised to produce the given number. For example, the logarithm of 8 with base 2 is 3, because $2^3 = 8$.

Q: Can I use a calculator to evaluate a logarithmic expression? A: Yes, you can use a calculator to evaluate a logarithmic expression. Most calculators have a logarithm button that allows you to enter the base and the value, and it will give you the logarithm.

Q: What is the difference between a common logarithm and a natural logarithm? A: A common logarithm is a logarithm with base 10, while a natural logarithm is a logarithm with base e (approximately 2.718).

Q: How do I convert a common logarithm to a natural logarithm? A: To convert a common logarithm to a natural logarithm, you can use the following formula: $\log_b(x) = \frac{\ln(x)}{\ln(b)}$.

Q: How do I convert a natural logarithm to a common logarithm? A: To convert a natural logarithm to a common logarithm, you can use the following formula: $\ln(x) = \log_{10}(x) \cdot \ln(10)$.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and they play a crucial role in various fields, including science, engineering, and economics. We hope that this article has provided you with a comprehensive understanding of logarithmic expressions and their properties.

Additional Resources

For further reading on logarithmic expressions, we recommend the following resources:

• Wikipedia: Logarithm: A comprehensive article on logarithmic expressions, including their properties and applications.
• Khan Academy: Logarithms: A video tutorial on logarithmic expressions, including their properties and applications.
• Math Is Fun: Logarithms: A website that provides interactive lessons and exercises on logarithmic expressions.

Final Answer

The final answer is: $\boxed{A}$

FAQs

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: What is the product 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 factors.

Q: What is the quotient rule for logarithmic expressions? A: The quotient rule states that the logarithm of a quotient is equal to the difference of the logarithms of the individual factors.

Q: What is the power rule for logarithmic expressions? A: The power rule states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q: How do I evaluate a logarithmic expression? A: To evaluate a logarithmic expression, you need to find the value of the logarithm. This can be done using a calculator or by using the properties of logarithms.

Q: What is the base of a logarithmic expression? A: The base of a logarithmic expression is the number that is raised to a power to produce the given value. For example, in the expression $\log_2(x)$, the base is 2.

Q: What is the logarithm of a number? A: The logarithm of a number is the power to which the base must be raised to produce the given number. For example, the logarithm of 8 with base 2 is 3, because $2^3 = 8$.