Which Of These Expressions Is Equivalent To $\log \left(9^2\right$\]?A. $2 \cdot \log (9$\] B. $\log (2) - \log (9$\] C. $\log (2) + \log (9$\] D. $\log (2) \cdot \log (9$\]

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. In this article, we will delve into the world of logarithms and explore which expression is equivalent to $\log \left(9^2\right)$. We will analyze each option carefully and provide a detailed explanation of the correct answer.

Logarithmic Properties

Before we dive into the problem, let's review some essential logarithmic properties:

• Product Rule: $\log (a \cdot b) = \log (a) + \log (b)$
• Power Rule: $\log (a^b) = b \cdot \log (a)$
• Quotient Rule: $\log \left(\frac{a}{b}\right) = \log (a) - \log (b)$

These properties will be instrumental in solving the problem at hand.

The Problem

We are given the expression $\log \left(9^2\right)$ and need to find an equivalent expression among the options provided.

Option A: $2 \cdot \log (9)$

Using the Power Rule, we can rewrite the expression as:

$$\log \left(9^2\right) = 2 \cdot \log (9)$$

This option seems promising, but let's analyze the other options as well.

Option B: $\log (2) - \log (9)$

Using the Quotient Rule, we can rewrite the expression as:

$$\log \left(9^2\right) = \log (9^2) = \log (9) + \log (9) = 2 \cdot \log (9)$$

However, this option introduces an unnecessary $\log (2)$ term, which is not present in the original expression.

Option C: $\log (2) + \log (9)$

Using the Product Rule, we can rewrite the expression as:

$$\log \left(9^2\right) = \log (9) + \log (9) = 2 \cdot \log (9)$$

This option also seems promising, but let's analyze the final option as well.

Option D: $\log (2) \cdot \log (9)$

This option is not equivalent to the original expression, as it introduces an unnecessary multiplication of the logarithms.

Conclusion

After analyzing each option carefully, we can conclude that the correct answer is:

Option A: $2 \cdot \log (9)$

This option is equivalent to the original expression $\log \left(9^2\right)$, as it correctly applies the Power Rule to simplify the expression.

Final Thoughts

In conclusion, understanding logarithmic properties is crucial for solving mathematical problems. By applying the Power Rule, we can simplify the expression $\log \left(9^2\right)$ to $2 \cdot \log (9)$. This example demonstrates the importance of logarithmic properties in mathematics and highlights the need for careful analysis when solving mathematical problems.

Additional Resources

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

• Khan Academy: Logarithms
• Math Is Fun: Logarithms
• Wolfram MathWorld: Logarithm

Introduction

In our previous article, we explored the concept of logarithmic expressions and analyzed which expression is equivalent to $\log \left(9^2\right)$. In this article, we will provide a comprehensive Q&A guide to help you better understand logarithmic expressions and their properties.

Q: What is a logarithmic expression?

A: A logarithmic expression is an expression that involves a logarithm, which is the inverse operation of exponentiation. In other words, a logarithmic expression is a way of expressing a number in terms of its logarithm.

Q: What are the basic properties of logarithms?

A: The basic properties of logarithms are:

• Product Rule: $\log (a \cdot b) = \log (a) + \log (b)$
• Power Rule: $\log (a^b) = b \cdot \log (a)$
• Quotient Rule: $\log \left(\frac{a}{b}\right) = \log (a) - \log (b)$

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the following steps:

1. Identify the logarithmic expression and the properties that apply to it.
2. Apply the properties to simplify the expression.
3. Use the simplified expression to solve the problem.

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

A: A logarithmic expression is an expression that involves a logarithm, while an exponential expression is an expression that involves an exponent. In other words, a logarithmic expression is a way of expressing a number in terms of its logarithm, while an exponential expression is a way of expressing a number in terms of its exponent.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the following steps:

1. Identify the logarithmic expression and the base of the logarithm.
2. Use the properties of logarithms to simplify the expression.
3. Use the simplified expression to evaluate the logarithm.

Q: What are some common logarithmic expressions?

A: Some common logarithmic expressions include:

• $\log (a)$
• $\log (a^b)$
• $\log \left(\frac{a}{b}\right)$
• $\log (a \cdot b)$

Q: How do I use logarithmic expressions in real-world problems?

A: Logarithmic expressions are used in a variety of real-world problems, 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 concentration of a substance.
• Engineering: Logarithmic expressions are used to calculate the power of a signal and the frequency of a wave.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding their properties is crucial for solving various mathematical problems. By applying the properties of logarithms, we can simplify and evaluate logarithmic expressions, and use them to solve real-world problems.

Additional Resources

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

• Khan Academy: Logarithms
• Math Is Fun: Logarithms
• Wolfram MathWorld: Logarithm

These resources provide a comprehensive introduction to logarithmic expressions and offer additional examples and exercises to practice.