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. 20 ⋅ Log ⁡ ( 27 20 \cdot \log (27 20 ⋅ Lo G ( 27 ]C. Log ⁡ ( 20 ) ⋅ Log ⁡ ( 27 \log (20) \cdot \log (27 Lo G ( 20 ) ⋅ Lo G ( 27 ]D. Log ⁡ ( 20 ) − Log ⁡ ( 27 \log (20) - \log (27 Lo G ( 20 ) − Lo G ( 27 ]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, used to represent the power to which a base number must be raised to produce a given value. In this article, we will delve into the world of logarithms and explore which of the given expressions is equivalent to $\log (20 \cdot 27)$.

The Basics of Logarithms

Before we dive into the analysis, let's briefly review the basics of logarithms. A logarithm is the inverse operation of exponentiation. In other words, if $x = a^y$, then $y = \log_a(x)$. The logarithm of a number to a certain base is the exponent to which the base must be raised to produce that number.

The Product Rule of Logarithms

One of the most important properties of logarithms is the product rule, which states that $\log(ab) = \log(a) + \log(b)$. This rule allows us to simplify complex logarithmic expressions by breaking them down into simpler components.

The Given Expressions

Now, let's examine the given expressions and determine which one is equivalent to $\log (20 \cdot 27)$.

A. $\log (20) + \log (27)$

This expression is based on the product rule of logarithms, which states that $\log(ab) = \log(a) + \log(b)$. Therefore, $\log (20 \cdot 27) = \log (20) + \log (27)$.

B. $20 \cdot \log (27)$

This expression is not equivalent to $\log (20 \cdot 27)$. The product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$, not $\log(a) \cdot \log(b)$.

C. $\log (20) \cdot \log (27)$

This expression is also not equivalent to $\log (20 \cdot 27)$. As mentioned earlier, the product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$, not $\log(a) \cdot \log(b)$.

D. $\log (20) - \log (27)$

This expression is not equivalent to $\log (20 \cdot 27)$. The product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$, not $\log(a) - \log(b)$.

Conclusion

Based on the analysis, we can conclude that the expression $\log (20) + \log (27)$ is equivalent to $\log (20 \cdot 27)$. This is because the product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$, which is the basis for this 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 calculate the pH of a solution and the intensity of a sound wave.
• Engineering: Logarithmic expressions are used to calculate the gain of an amplifier and the frequency response of a filter.

Final Thoughts

Introduction

In our previous article, we explored the concept of logarithmic expressions and determined which of the given expressions is equivalent to $\log (20 \cdot 27)$. In this article, we will delve deeper into the world of logarithms and answer some of the most frequently asked questions about logarithmic expressions.

Q&A

Q: What is the product rule of logarithms?

A: The product rule of logarithms states that $\log(ab) = \log(a) + \log(b)$. This rule allows us to simplify complex logarithmic expressions by breaking them down into simpler components.

Q: How do I apply the product rule of logarithms?

A: To apply the product rule of logarithms, simply break down the logarithmic expression into its individual components and add the logarithms of each component. For example, $\log(20 \cdot 27) = \log(20) + \log(27)$.

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

A: A logarithmic expression is the inverse of an exponential expression. In other words, if $x = a^y$, then $y = \log_a(x)$. Exponential expressions are used to represent the power to which a base number must be raised to produce a given value.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, apply the product rule of logarithms and any other relevant logarithmic properties. For example, $\log(20 \cdot 27) = \log(20) + \log(27) = \log(20) + \log(3^3) = \log(20) + 3\log(3)$.

Q: What is the logarithmic property of addition?

A: The logarithmic property of addition states that $\log(a + b) = \log(a) + \log(1 + \frac{b}{a})$. This property allows us to simplify complex logarithmic expressions involving addition.

Q: How do I apply the logarithmic property of addition?

A: To apply the logarithmic property of addition, simply break down the logarithmic expression into its individual components and add the logarithms of each component. For example, $\log(20 + 27) = \log(20) + \log(1 + \frac{27}{20}) = \log(20) + \log(1.35)$.

Q: What is the logarithmic property of subtraction?

A: The logarithmic property of subtraction states that $\log(a - b) = \log(a) - \log(1 - \frac{b}{a})$. This property allows us to simplify complex logarithmic expressions involving subtraction.

Q: How do I apply the logarithmic property of subtraction?

A: To apply the logarithmic property of subtraction, simply break down the logarithmic expression into its individual components and subtract the logarithms of each component. For example, $\log(20 - 27) = \log(20) - \log(1 - \frac{27}{20}) = \log(20) - \log(0.85)$.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, used to represent the power to which a base number be raised to produce a given value. By understanding the product rule of logarithms and other logarithmic properties, we can simplify complex logarithmic expressions and answer some of the most frequently asked questions about logarithmic expressions.

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 calculate the pH of a solution and the intensity of a sound wave.
• Engineering: Logarithmic expressions are used to calculate the gain of an amplifier and the frequency response of a filter.

Final Thoughts

In conclusion, logarithmic expressions are a powerful tool in mathematics, used to represent the power to which a base number must be raised to produce a given value. By understanding the product rule of logarithms and other logarithmic properties, we can simplify complex logarithmic expressions and answer some of the most frequently asked questions about logarithmic expressions.