Evaluate The Following Expression. Round Your Answer To Two Decimal Places. $\log _7 E$A. 1.47 B. 0.51 C. 0.43 D. 1.95

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 $\log _7 e$. We will break down the solution into manageable steps, and by the end of this article, you will have a clear understanding of how to evaluate this expression.

Understanding Logarithmic Expressions

Before we dive into the solution, let's take a moment to understand what logarithmic expressions are. A logarithmic expression is a mathematical operation that represents the power to which a base number must be raised to produce a given value. In other words, if we have a logarithmic expression $\log _b a$, it means that we are looking for the exponent to which the base $b$ must be raised to produce the value $a$.

Evaluating the Expression $\log _7 e$

Now that we have a basic understanding of logarithmic expressions, let's focus on evaluating the expression $\log _7 e$. To do this, we need to use 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 the change of base formula, we can rewrite the expression $\log _7 e$ as follows:

$$\log _7 e = \frac{\log _e e}{\log _e 7}$$

Simplifying the Expression

Now that we have rewritten the expression, let's simplify it further. We know that $\log _e e = 1$, since the logarithm of a number to its own base is always 1. Therefore, we can simplify the expression as follows:

$$\log _7 e = \frac{1}{\log _e 7}$$

Finding the Value of $\log _e 7$

To find the value of $\log _e 7$, we can use a calculator or a logarithmic table. However, for the sake of this article, let's assume that we don't have access to a calculator or a logarithmic table. In this case, we can use the fact that $\log _e 7 = \frac{\ln 7}{\ln e}$, where $\ln$ represents the natural logarithm.

Since $\ln e = 1$, we can simplify the expression as follows:

$$\log _e 7 = \frac{\ln 7}{1} = \ln 7$$

Evaluating the Expression

Now that we have found the value of $\log _e 7$, we can evaluate the expression $\log _7 e$ as follows:

$$\log _7 e = \frac{1}{\log _e 7} = \frac{1}{\ln 7}$$

Using a Calculator to Find the Value

To find the value of $\frac{1}{\ln 7}$, we can use a calculator. Plugging in the value of $\ln 7$ into a calculator, we get:

$$\frac{1}{\ln 7} \approx 0.43$$

Conclusion

In this article, we evaluated the expression $\log _7 e$ using the change of base formula and the properties of logarithms. We simplified the expression and found its value using a calculator. The final answer is $\boxed{0.43}$.

Discussion

The expression $\log _7 e$ is a fundamental concept in mathematics, and understanding how to evaluate it is crucial for solving various mathematical problems. In this article, we provided a step-by-step guide on how to evaluate this expression, and by the end of this article, you should have a clear understanding of how to evaluate logarithmic expressions.

Practice Problems

To practice what you have learned, try evaluating the following expressions:

1. $\log _5 2$
2. $\log _3 4$
3. $\log _2 8$

Use the change of base formula and the properties of logarithms to simplify and evaluate these expressions.

References

• [1] "Logarithmic Expressions" by Math Open Reference
• [2] "Change of Base Formula" by Khan Academy
• [3] "Properties of Logarithms" by Wolfram MathWorld

Additional Resources

For more information on logarithmic expressions and how to evaluate them, check out the following resources:

• [1] "Logarithmic Expressions" by Mathway
• [2] "Change of Base Formula" by Purplemath
• [3] "Properties of Logarithms" by IXL Math
  Logarithmic Expressions Q&A =============================

Introduction

In our previous article, we evaluated the expression $\log _7 e$ using the change of base formula and the properties of logarithms. In this article, we will provide a Q&A section to help you better understand logarithmic expressions and how to evaluate them.

Q&A

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 produce a given value.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithmic expression. It is given by the formula $\log _b a = \frac{\log _c a}{\log _c b}$, where $c$ is any positive real number.

Q: How do I evaluate a logarithmic expression?

A: To evaluate a logarithmic expression, you can use the change of base formula and the properties of logarithms. You can also use a calculator or a logarithmic table to find the value of the expression.

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: 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 always a good idea to understand the underlying mathematics behind the expression before using a calculator.

Q: How do I simplify a logarithmic expression?

A: To simplify a logarithmic expression, you can use the properties of logarithms, such as the product rule and the quotient rule. You can also use the change of base formula to change the base of the expression.

Q: What is the value of $\log _e 7$?

A: The value of $\log _e 7$ is approximately 1.9459.

Q: How do I evaluate the expression $\log _7 e$?

A: To evaluate the expression $\log _7 e$, you can use the change of base formula and the properties of logarithms. You can also use a calculator or a logarithmic table to find the value of the expression.

Q: What is the final answer to the expression $\log _7 e$?

A: The final answer to the expression $\log _7 e$ is approximately 0.43.

Practice Problems

To practice what you have learned, try evaluating the following expressions:

1. $\log _5 2$
2. $\log _3 4$
3. $\log _2 8$

Use the change of base formula and the properties of logarithms to simplify and evaluate these expressions.

References

• [1] "Logarithmic Expressions" by Math Open Reference
• [2] "Change of Base Formula" by Khan Academy
• [3] "Properties of Logarithms" by Wolfram MathWorld

Additional Resources

For more information on logarithmic expressions and how to evaluate them, check out the following resources:

• [1] "Logarithmic Expressions" by Mathway
• [2] "Change of Base Formula" by Purplemath
• [3] "Properties of Logarithms" by IXL Math

Conclusion

In this article, we provided a Q&A section to help you better understand logarithmic expressions and how to evaluate them. We covered topics such as the change of base formula, the properties of logarithms, and how to simplify and evaluate logarithmic expressions. We also provided practice problems and additional resources for further learning.