Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation?$\[ \log _2 X=24 \\]A. \[$x^2=24\$\]B. \[$x^{24}=2\$\]C. \[$2^x=24\$\]D. \[$2^{24}=x\$\]

Introduction

Exponential and logarithmic equations are fundamental concepts in mathematics, and understanding how to solve them is crucial for success in various fields, including science, engineering, and economics. In this article, we will focus on solving exponential equations that are equivalent to logarithmic equations. We will use the given logarithmic equation $\log_2 x = 24$ as an example and explore the different options provided.

Understanding Logarithmic Equations

A logarithmic equation is an equation that involves a logarithm, which is the inverse operation of exponentiation. The logarithmic equation $\log_2 x = 24$ can be rewritten in exponential form as $2^{24} = x$. This means that the logarithmic equation is asking for the value of $x$ that satisfies the equation $2^{24} = x$.

Option A: $x^2 = 24$

Option A is $x^2 = 24$. This equation is not equivalent to the logarithmic equation $\log_2 x = 24$. To see why, let's consider the properties of logarithms. The logarithmic equation $\log_2 x = 24$ implies that $x$ is equal to $2^{24}$. However, $x^2 = 24$ does not imply that $x$ is equal to $2^{24}$. In fact, $x^2 = 24$ has two solutions: $x = \sqrt{24}$ and $x = -\sqrt{24}$. Therefore, option A is not the correct answer.

Option B: $x^{24} = 2$

Option B is $x^{24} = 2$. This equation is not equivalent to the logarithmic equation $\log_2 x = 24$. To see why, let's consider the properties of logarithms. The logarithmic equation $\log_2 x = 24$ implies that $x$ is equal to $2^{24}$. However, $x^{24} = 2$ implies that $x$ is equal to $2^{1/24}$. Therefore, option B is not the correct answer.

Option C: $2^x = 24$

Option C is $2^x = 24$. This equation is not equivalent to the logarithmic equation $\log_2 x = 24$. To see why, let's consider the properties of logarithms. The logarithmic equation $\log_2 x = 24$ implies that $x$ is equal to $2^{24}$. However, $2^x = 24$ implies that $x$ is equal to $\log_2 24$. Therefore, option C is not the correct answer.

Option D: $2^{24} = x$

Option D is $2^{24} = x$. This equation is equivalent to the logarithmic equation $\log_2 x = 24$. To see why, let's consider the properties of logarithms. The logarithmic equation $\log_2 x = 24$ implies that $x$ is equal to $2^{24}$. Therefore, option D is the correct answer.

Conclusion

In conclusion, the correct answer is option D: $2^{24} = x$. This equation is equivalent to the logarithmic equation $\log_2 x = 24$. We hope that this article has provided a clear understanding of how to solve exponential equations that are equivalent to logarithmic equations.

Key Takeaways

• Exponential and logarithmic equations are fundamental concepts in mathematics.
• Understanding how to solve exponential equations that are equivalent to logarithmic equations is crucial for success in various fields.
• The logarithmic equation $\log_2 x = 24$ can be rewritten in exponential form as $2^{24} = x$.
• Option D: $2^{24} = x$ is the correct answer.

Final Thoughts

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

A: An exponential equation is an equation that involves an exponent, which is a power or a root. For example, $x^2 = 4$ is an exponential equation. A logarithmic equation, on the other hand, is an equation that involves a logarithm, which is the inverse operation of exponentiation. For example, $\log_2 x = 2$ is a logarithmic equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable (x) by getting rid of the exponent. You can do this by using the following steps:

1. Take the logarithm of both sides of the equation.
2. Use the property of logarithms to bring the exponent down.
3. Simplify the equation and solve for x.

Q: How do I solve a logarithmic equation?

A: To solve a logarithmic equation, you need to isolate the variable (x) by getting rid of the logarithm. You can do this by using the following steps:

1. Exponentiate both sides of the equation.
2. Use the property of exponentiation to bring the logarithm up.
3. Simplify the equation and solve for x.

Q: What is the relationship between exponential and logarithmic equations?

A: Exponential and logarithmic equations are inverse operations. This means that if you have an exponential equation, you can convert it to a logarithmic equation by taking the logarithm of both sides. Similarly, if you have a logarithmic equation, you can convert it to an exponential equation by exponentiating both sides.

Q: How do I choose between an exponential equation and a logarithmic equation?

A: When choosing between an exponential equation and a logarithmic equation, you need to consider the context of the problem. If the problem involves growth or decay, an exponential equation is often the best choice. If the problem involves a relationship between two quantities, a logarithmic equation may be more suitable.

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

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

• Not checking the domain of the logarithm.
• Not using the correct base for the logarithm.
• Not simplifying the equation before solving for x.
• Not checking the solution to make sure it is valid.

Q: How can I practice solving exponential and logarithmic equations?

A: There are many ways to practice solving exponential and logarithmic equations, including:

• Working through practice problems in a textbook or online resource.
• Using online calculators or software to solve equations.
• Creating your own practice problems and solving them.
• Joining a study group or working with a tutor to practice solving equations.

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

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

• Modeling population growth and decay.
• Calculating interest rates and investments.
• Analyzing data and trends.
• Solving problems in physics, engineering, and other fields.

Conclusion

In conclusion, exponential and logarithmic equations are fundamental concepts in mathematics that have many real-world applications. By understanding how to solve these equations, you can develop the skills and confidence needed to tackle complex problems and achieve success in your mathematical pursuits. Remember to always check your work and verify your answers to ensure accuracy and precision. With practice and patience, you can become proficient in solving exponential and logarithmic equations and achieve your goals.