Select The Correct Answer.Which Exponential Equation Is Equivalent To This Logarithmic Equation? Log ⁡ X 5 + Log ⁡ X 12 = 7 \log_x 5 + \log_x 12 = 7 Lo G X ​ 5 + Lo G X ​ 12 = 7 A. 7 X = 60 7^x = 60 7 X = 60 B. X 7 = 60 X^7 = 60 X 7 = 60 C. X 7 = 17 X^7 = 17 X 7 = 17 D. 7 X = 17 7^x = 17 7 X = 17

Introduction

Logarithmic and exponential 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 $\log_x 5 + \log_x 12 = 7$ 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 $\log_x 5 + \log_x 12 = 7$ can be rewritten using the properties of logarithms as $\log_x (5 \cdot 12) = 7$. This simplifies to $\log_x 60 = 7$.

Converting Logarithmic Equations to Exponential Equations

To convert a logarithmic equation to an exponential equation, we can use the definition of a logarithm. The logarithmic equation $\log_x 60 = 7$ can be rewritten as $x^7 = 60$. This is because the logarithm of a number to a certain base is equal to the exponent to which the base must be raised to produce that number.

Analyzing the Options

Now that we have converted the logarithmic equation to an exponential equation, we can analyze the options provided.

Option A: $7^x = 60$

This option is incorrect because the base of the exponential equation is 7, not x. We are looking for an exponential equation with x as the base.

Option B: $x^7 = 60$

This option is correct because it matches the exponential equation we derived from the logarithmic equation.

Option C: $x^7 = 17$

This option is incorrect because the right-hand side of the equation is 17, not 60.

Option D: $7^x = 17$

This option is incorrect because the base of the exponential equation is 7, not x, and the right-hand side of the equation is 17, not 60.

Conclusion

In conclusion, the correct answer is Option B: $x^7 = 60$. This is because it matches the exponential equation we derived from the logarithmic equation. Understanding how to solve exponential and logarithmic equations is crucial for success in various fields, and this article has provided a step-by-step guide on how to solve these types of equations.

Additional Tips and Tricks

• When converting a logarithmic equation to an exponential equation, make sure to use the definition of a logarithm.
• When analyzing the options, make sure to check the base and the right-hand side of the equation.
• Practice solving exponential and logarithmic equations to become proficient in solving these types of equations.

Common Mistakes to Avoid

• Not using the definition of a logarithm when converting a logarithmic equation to an exponential equation.
• Not checking the base and the right-hand side of the equation when analyzing the options.
• Not practicing solving exponential and logarithmic equations to become proficient in solving these types of equations.

Real-World Applications

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

• Finance: Exponential and logarithmic equations are used to calculate interest rates, investment returns, and stock prices.
• Science: Exponential and logarithmic equations are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Exponential and logarithmic equations are used to design and optimize systems, including electrical circuits, mechanical systems, and computer networks.

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 to which a number is raised. A logarithmic equation, on the other hand, is an equation that involves a logarithm, which is the inverse operation of exponentiation.

Q: How do I convert a logarithmic equation to an exponential equation?

A: To convert a logarithmic equation to an exponential equation, you can use the definition of a logarithm. The logarithmic equation $\log_x 60 = 7$ can be rewritten as $x^7 = 60$.

Q: What is the base of an exponential equation?

A: The base of an exponential equation is the number that is raised to a power. In the equation $x^7 = 60$, the base is x.

Q: What is the right-hand side of an exponential equation?

A: The right-hand side of an exponential equation is the number that the base is raised to. In the equation $x^7 = 60$, the right-hand side is 60.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the definition of an exponential function. For example, to solve the equation $x^7 = 60$, you can take the seventh root of both sides to get $x = \sqrt[7]{60}$.

Q: What is the inverse operation of exponentiation?

A: The inverse operation of exponentiation is logarithm. This means that if you have an exponential equation, you can convert it to a logarithmic equation by taking the logarithm of both sides.

Q: How do I use logarithms to solve exponential equations?

A: To use logarithms to solve exponential equations, you can take the logarithm of both sides of the equation. For example, to solve the equation $x^7 = 60$, you can take the logarithm of both sides to get $\log_x 60 = 7$.

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 using the definition of an exponential function when solving exponential equations.
• Not using the definition of a logarithm when converting a logarithmic equation to an exponential equation.
• Not checking the base and the right-hand side of the equation when analyzing the options.
• Not practicing solving exponential and logarithmic equations to become proficient in solving these types of equations.

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

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

• Finance: Exponential and logarithmic equations are used to calculate interest rates, investment returns, and stock prices.
• Science: Exponential and logarithmic equations are used to model population growth, chemical reactions, and physical phenomena.
• Engineering: Exponential and logarithmic equations are used to design and optimize systems, including electrical circuits, mechanical systems, and computer networks.

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

A: You can practice solving exponential and logarithmic equations by working through practice problems and exercises. You can also use online resources, such as calculators and software, to help you solve these types of equations.

Q: What are some tips for becoming proficient in solving exponential and logarithmic equations?

A: Some tips for becoming proficient in solving exponential and logarithmic equations include:

• Practicing regularly to build your skills and confidence.
• Using online resources, such as calculators and software, to help you solve these types of equations.
• Working through practice problems and exercises to build your problem-solving skills.
• Seeking help from a teacher or tutor if you are struggling with these types of equations.