Choose The Best Answer.$ 7^{\log _7 8} = $A. 8 B. 7 C. $ 7^8 $ D. $ 8^7 $

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of logarithms and exponents. In this article, we will explore how to solve exponential equations, with a focus on the specific problem of $7^{\log _7 8}$. We will break down the solution step by step, using clear and concise language to ensure that readers understand the underlying concepts.

Understanding Exponents and Logarithms

Before we dive into the solution, let's review the basics of exponents and logarithms.

• Exponents: An exponent is a small number that is raised to a power. For example, $2^3$ means $2$ multiplied by itself $3$ times, or $2 \times 2 \times 2 = 8$.
• Logarithms: A logarithm is the inverse operation of an exponent. For example, if we have $2^3 = 8$, then the logarithm of $8$ with base $2$ is $3$, denoted as $\log_2 8 = 3$.

Solving the Equation

Now that we have reviewed the basics of exponents and logarithms, let's tackle the equation $7^{\log _7 8}$.

To solve this equation, we need to use the property of logarithms that states $\log_a a = 1$. This means that if we have $\log_7 7$, the answer is $1$.

Using this property, we can rewrite the equation as:

$$7^{\log _7 8} = 7^1$$

Now, we can simplify the right-hand side of the equation by evaluating $7^1$, which is equal to $7$.

Therefore, the solution to the equation $7^{\log _7 8}$ is:

$$7^{\log _7 8} = 7$$

Conclusion

In this article, we have solved the exponential equation $7^{\log _7 8}$ using the properties of logarithms and exponents. We have shown that the solution to this equation is $7$, which is the correct answer.

Choosing the Best Answer

Now that we have solved the equation, let's review the answer choices:

A. 8 B. 7 C. $7^8$ D. $8^7$

Based on our solution, we can see that the correct answer is:

B. 7

Therefore, the best answer is B. 7.

Final Thoughts

$$$$

Introduction

In our previous article, we explored how to solve exponential equations, with a focus on the specific problem of $7^{\log _7 8}$. We broke down the solution step by step, using clear and concise language to ensure that readers understand the underlying concepts.

In this article, we will continue to explore the topic of exponential equations, this time in the form of a Q&A guide. We will answer common questions and provide additional examples to help readers solidify their understanding of this important mathematical concept.

Q&A

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponent, which is a small number that is raised to a power. For example, $2^3$ is an exponential equation because it involves the exponent $3$.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to use the properties of logarithms and exponents. You can start by rewriting the equation in a simpler form, and then use the properties of logarithms to simplify the equation further.

Q: What is the property of logarithms that states $\log_a a = 1$?

A: This property states that if we have $\log_a a$, the answer is $1$. This means that if we have $\log_7 7$, the answer is $1$.

Q: How do I use the property of logarithms to simplify an exponential equation?

A: To use the property of logarithms to simplify an exponential equation, you need to rewrite the equation in a form that involves a logarithm. For example, if we have $7^{\log _7 8}$, we can rewrite it as $7^1$.

Q: What is the solution to the equation $7^{\log _7 8}$?

A: The solution to the equation $7^{\log _7 8}$ is $7$.

Q: What are some common mistakes to avoid when solving exponential equations?

A: Some common mistakes to avoid when solving exponential equations include:

• Not using the properties of logarithms and exponents correctly
• Not rewriting the equation in a simpler form
• Not checking the solution to make sure it is correct

Q: How do I check my solution to an exponential equation?

A: To check your solution to an exponential equation, you need to plug it back into the original equation and make sure it is true. For example, if we have the solution $7$ to the equation $7^{\log _7 8}$, we can plug it back into the original equation to make sure it is true.

Q: What are some real-world applications of exponential equations?

A: Exponential equations have many real-world applications, including:

• Modeling population growth
• Modeling financial growth
• Modeling chemical reactions

Q: How do I use exponential equations to model real-world problems?

A: To use exponential equations to model real-world problems, you need to identify the variables and the relationships between them. You can then use the properties of logarithms and exponents to simplify the equation and arrive at a solution.

Conclusion

In this article, we have provided a Q&A guide to help readers understand exponential equations. We have answered common questions and provided additional examples to help readers solidify their understanding of this important mathematical concept.

Final Thoughts

Exponential equations are a fundamental concept in mathematics, and they have many real-world applications. By understanding how to solve exponential equations, you can model complex problems and arrive at accurate solutions. We hope that this Q&A guide has been helpful in your understanding of exponential equations.

Additional Resources

For additional resources on exponential equations, including videos, tutorials, and practice problems, please visit the following websites:

• Khan Academy: Exponential Equations
• Mathway: Exponential Equations
• Wolfram Alpha: Exponential Equations

Practice Problems

To practice solving exponential equations, try the following problems:

1. Solve the equation $2^{\log _2 3}$
2. Solve the equation $3^{\log _3 4}$
3. Solve the equation $5^{\log _5 6}$

We hope that this Q&A guide has been helpful in your understanding of exponential equations. If you have any further questions or need additional help, please don't hesitate to ask.