Evaluate The Following Expression. Round Your Answer To Two Decimal Places. $\log_8 E$A. 2.08 B. 1.75 C. 0.48 D. 0.43

Introduction

Logarithmic expressions are a fundamental concept in mathematics, and understanding how to evaluate them is crucial for solving various mathematical problems. In this article, we will focus on evaluating the expression $\log_8 e$ and provide a step-by-step guide on how to solve it.

Understanding Logarithmic Expressions

A logarithmic expression is a mathematical operation that represents the power to which a base number must be raised to obtain a given value. In other words, it is the inverse operation of exponentiation. The general form of a logarithmic expression is $\log_b a = c$, where $b$ is the base, $a$ is the argument, and $c$ is the result.

Evaluating the Expression $\log_8 e$

To evaluate the expression $\log_8 e$, we need to understand the properties of logarithms. One of the key properties is the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

Using this property, we can rewrite the expression $\log_8 e$ as $\frac{\log e}{\log 8}$. Now, we need to find the values of $\log e$ and $\log 8$.

Finding the Value of $\log e$

The value of $\log e$ is a fundamental constant in mathematics, and it is approximately equal to 1. The exact value of $\log e$ is not a rational number, but it can be approximated using various mathematical techniques.

Finding the Value of $\log 8$

To find the value of $\log 8$, we can use the change of base formula. Let's choose base 2 as our new base. Then, we have $\log 8 = \frac{\log 2^3}{\log 2} = \frac{3 \log 2}{\log 2} = 3$.

Evaluating the Expression

Now that we have found the values of $\log e$ and $\log 8$, we can evaluate the expression $\log_8 e$. We have:

$\log_8 e = \frac{\log e}{\log 8} = \frac{1}{3}$

Rounding the Answer

The problem asks us to round our answer to two decimal places. Therefore, we need to round $\frac{1}{3}$ to two decimal places.

$\frac{1}{3} \approx 0.33$

Conclusion

In this article, we evaluated the expression $\log_8 e$ using the change of base formula and found that the answer is approximately equal to 0.33. We also discussed the properties of logarithms and how to use them to solve various mathematical problems.

Answer

The correct answer is 0.33.

Comparison with Other Options

Let's compare our answer with the other options:

• A. 2.08: This is not a correct answer.
• B. 1.75: This is not a correct answer.
• C. 0.48: This is not a correct answer.
• D. 0.43: This is not a correct answer.

Our answer, 0.33, is the only correct option.

Final Thoughts

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Introduction

In our previous article, we evaluated the expression $\log_8 e$ and found that the answer is approximately equal to 0.33. In this article, we will provide a Q&A guide to help you understand logarithmic expressions and how to use them to solve various mathematical problems.

Q: What is a logarithmic expression?

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

Q: What is the change of base formula?

A: The change of base formula is a property of logarithms that states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive real number.

Q: How do I use the change of base formula to evaluate a logarithmic expression?

A: To use the change of base formula, you need to choose a new base and rewrite the expression using that base. Then, you can use the formula to evaluate the expression.

Q: What is the value of $\log e$?

A: The value of $\log e$ is a fundamental constant in mathematics, and it is approximately equal to 1.

Q: How do I find the value of $\log 8$?

A: To find the value of $\log 8$, you can use the change of base formula. Let's choose base 2 as our new base. Then, we have $\log 8 = \frac{\log 2^3}{\log 2} = \frac{3 \log 2}{\log 2} = 3$.

Q: How do I evaluate the expression $\log_8 e$?

A: To evaluate the expression $\log_8 e$, you need to use the change of base formula. We have:

$\log_8 e = \frac{\log e}{\log 8} = \frac{1}{3}$

Q: How do I round my answer to two decimal places?

A: To round your answer to two decimal places, you need to look at the third decimal place. If it is less than 5, you round down. If it is 5 or greater, you round up.

Q: What is the correct answer to the expression $\log_8 e$?

A: The correct answer is approximately equal to 0.33.

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

A: Some common mistakes to avoid when evaluating logarithmic expressions include:

• Not using the change of base formula
• Not choosing a new base
• Not rewriting the expression using the new base
• Not using the correct formula to evaluate the expression

Q: How can I practice evaluating logarithmic expressions?

A: You can practice evaluating logarithmic expressions by working through examples and exercises. You can also use online resources and calculators to help you evaluate expressions.

Conclusion

In this article, we provided a Q&A guide to help you understand logarithmic expressions and how to use them to solve various mathematical problems. We hope that this article has been helpful in understanding logarithmic expressions and how to use them to solve various mathematical problems.

Additional Resources

• Logarithmic Expressions Tutorial
• Change of Base Formula
• Logarithmic Expressions Calculator

Final Thoughts

Evaluating logarithmic expressions is a crucial skill in mathematics, and it requires a deep understanding of the properties of logarithms. In this article, we provided a Q&A guide to help you understand logarithmic expressions and how to use them to solve various mathematical problems. We hope that this article has been helpful in understanding logarithmic expressions and how to use them to solve various mathematical problems.