Which Of These Expressions Is Equivalent To $\log (4^6$\]?A. $6 \cdot \log (4$\] B. $\log (6) - \log (4$\] C. $\log (6) \cdot \log (4$\] D. $\log (6) + \log (4$\]

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 options is equivalent to $\log (4^6)$.

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, $x$ is the value, and $y$ is the logarithm.

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: $\log_b(x^y) = y \cdot \log_b(x)$

Evaluating the Given Expression

We are given the expression $\log (4^6)$. To evaluate this expression, we can use the power rule of logarithms, which states that $\log_b(x^y) = y \cdot \log_b(x)$.

Using this rule, we can rewrite the given expression as:

$$\log (4^6) = 6 \cdot \log (4)$$

Comparing with the Options

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

A. $6 \cdot \log (4)$ B. $\log (6) - \log (4)$ C. $\log (6) \cdot \log (4)$ D. $\log (6) + \log (4)$

From our evaluation, we can see that option A is equivalent to $\log (4^6)$.

Conclusion

In conclusion, the correct answer is option A: $6 \cdot \log (4)$. This is because the power rule of logarithms states that $\log_b(x^y) = y \cdot \log_b(x)$, and we can apply this rule to evaluate the given expression.

Final Thoughts

Understanding logarithmic expressions is crucial for solving various mathematical problems. By applying the properties of logarithmic expressions, we can simplify complex expressions and arrive at the correct solution. In this article, we have explored the concept of logarithmic expressions and determined which of the given options is equivalent to $\log (4^6)$. We hope that this article has provided valuable insights into the world of logarithmic expressions.

Additional Resources

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

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

Frequently Asked Questions

Q: What is the definition of a logarithmic expression? A: A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to produce a given value.

Q: What are the properties of logarithmic expressions? A: The properties of logarithmic expressions include the product rule, quotient rule, and power rule.

Q: How do I evaluate a logarithmic expression? A: To evaluate a logarithmic expression, you can use the power rule of logarithms, which states that $\log_b(x^y) = y \cdot \log_b(x)$.

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Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. In our previous article, we explored the concept of logarithmic expressions and determined which of the given options is equivalent to $\log (4^6)$. In this article, we will address some of the most frequently asked questions related to logarithmic expressions.

Q&A Session

Q1: What is the definition of a logarithmic expression?

A: 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.

Q2: What are the properties of logarithmic expressions?

A: The properties of logarithmic expressions include:

• 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: $\log_b(x^y) = y \cdot \log_b(x)$

Q3: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the power rule of logarithms, which states that $\log_b(x^y) = y \cdot \log_b(x)$. You can also use the product rule and quotient rule to simplify the expression.

Q4: 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.

Q5: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you can use the properties of logarithmic expressions to isolate the variable. For example, if you have the equation $\log_b(x) = y$, you can rewrite it as $b^y = x$.

Q6: 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.

Q7: Can I use a calculator to evaluate a logarithmic expression?

A: Yes, you can use a calculator to evaluate a logarithmic expression. However, it's essential to understand the properties of logarithmic expressions to use the calculator effectively.

Q8: How do I graph a logarithmic function?

A: To graph a logarithmic function, you can use a graphing calculator or a graphing software. You can also use the properties of logarithmic expressions to determine the shape and behavior of the function.

Q9: What is the domain and range of a logarithmic function?

A: The domain of a logarithmic function is all positive real numbers, while the range is all real numbers.

Q10: Can I use logarithmic expressions in real-world applications?

A: Yes, logarithmic expressions are used in various real-world applications, such as finance, science, and engineering.

Conclusion

In conclusion, logarithmic expressions are a fundamental concept in mathematics, and understanding them is crucial for solving various mathematical problems. By addressing some of the most frequently asked questions related to logarithmic expressions, we hope to provide valuable insights into the world of logarithmic expressions.

Additional Resources

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

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

Frequently Asked Questions (FAQs)

Q: What is the definition of a logarithmic expression? A: A logarithmic expression is a mathematical expression that represents the power to which a base number must be raised to produce a given value.

Q: What are the properties of logarithmic expressions? A: The properties of logarithmic expressions include the product rule, quotient rule, and power rule.

Q: How do I evaluate a logarithmic expression? A: To evaluate a logarithmic expression, you can use the power rule of logarithms, which states that $\log_b(x^y) = y \cdot \log_b(x)$.

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.

Logarithmic Expression Practice Problems

1. Evaluate the expression $\log_2(16)$.
2. Simplify the expression $\log_3(9) + \log_3(3)$.
3. Solve the equation $\log_4(x) = 2$.
4. Graph the function $y = \log_2(x)$.
5. Find the domain and range of the function $y = \log_3(x)$.

Answer Key

1. $\log_2(16) = 4$
2. $\log_3(9) + \log_3(3) = 2 + 1 = 3$
3. $x = 4^2 = 16$
4. The graph of the function $y = \log_2(x)$ is a logarithmic curve that passes through the point $(1, 0)$.
5. The domain of the function $y = \log_3(x)$ is all positive real numbers, while the range is all real numbers.