Use The Equation $2,401=7^{6-2x}$. What Power Of 7 Is 2,401?A. 3 B. 4 C. 5

Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the underlying principles. In this article, we will explore how to solve exponential equations, with a focus on finding the power of 7 in the number 2,401. We will use the equation $2,401=7^{6-2x}$ to demonstrate the steps involved in solving this type of problem.

Understanding Exponential Equations

Exponential equations involve a variable in the exponent, which can be a constant or another variable. In the equation $2,401=7^{6-2x}$, the variable $x$ is in the exponent, and we need to find its value. Exponential equations can be solved using logarithms, which are a mathematical operation that allows us to find the power to which a base number must be raised to obtain a given value.

Using Logarithms to Solve Exponential Equations

To solve the equation $2,401=7^{6-2x}$, we can use logarithms to find the value of $x$. We can start by taking the logarithm of both sides of the equation:

$$\log(2,401) = \log(7^{6-2x})$$

Using the property of logarithms that states $\log(a^b) = b\log(a)$, we can rewrite the equation as:

$$\log(2,401) = (6-2x)\log(7)$$

Now, we can use the fact that $\log(2,401) = \log(7^4)$ to simplify the equation:

$$4\log(7) = (6-2x)\log(7)$$

Dividing both sides of the equation by $\log(7)$, we get:

$$4 = 6-2x$$

Subtracting 6 from both sides of the equation, we get:

$$-2 = -2x$$

Dividing both sides of the equation by -2, we get:

$$x = 1$$

Finding the Power of 7 in 2,401

Now that we have found the value of $x$, we can use it to find the power of 7 in 2,401. We can start by rewriting the equation $2,401=7^{6-2x}$ as:

$$2,401 = 7^{6-2(1)}$$

Simplifying the exponent, we get:

$$2,401 = 7^4$$

Therefore, the power of 7 in 2,401 is 4.

Conclusion

In this article, we have demonstrated how to solve exponential equations using logarithms. We have used the equation $2,401=7^{6-2x}$ to find the value of $x$ and then used it to find the power of 7 in 2,401. The power of 7 in 2,401 is 4.

Answer

The correct answer is B. 4.

Additional Resources

For more information on exponential equations and logarithms, please refer to the following resources:

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

References

• "Algebra and Trigonometry" by Michael Sullivan
• "Calculus" by Michael Spivak
• "Mathematics for Computer Science" by Eric Lehman and Tom Leighton
  Frequently Asked Questions: Exponential Equations and Logarithms ====================================================================

Q: What is an exponential equation?

A: An exponential equation is an equation that involves a variable in the exponent. For example, the equation $2,401=7^{6-2x}$ is an exponential equation because the variable $x$ is in the exponent.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use logarithms to find the value of the variable in the exponent. You can start by taking the logarithm of both sides of the equation and then using the properties of logarithms to simplify the equation.

Q: What is a logarithm?

A: A logarithm is a mathematical operation that allows you to find the power to which a base number must be raised to obtain a given value. For example, the logarithm of 100 to the base 10 is 2, because $10^2 = 100$.

Q: How do I use logarithms to solve an exponential equation?

A: To use logarithms to solve an exponential equation, you can start by taking the logarithm of both sides of the equation. Then, you can use the properties of logarithms to simplify the equation and find the value of the variable in the exponent.

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

A: A logarithmic equation is an equation that involves a variable as the exponent of a logarithm. For example, the equation $\log(x) = 2$ is a logarithmic equation. An exponential equation, on the other hand, is an equation that involves a variable in the exponent. For example, the equation $2,401=7^{6-2x}$ is an exponential equation.

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

A: To convert an exponential equation to a logarithmic equation, you can take the logarithm of both sides of the equation. This will allow you to rewrite the equation in logarithmic form.

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 correct properties of logarithms
• Not simplifying the equation correctly
• Not checking the solution to make sure it is valid

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

A: To check your solution to an exponential equation, you can plug the value of the variable back into the original equation and make sure it is true. You can also use a calculator or computer program to check your solution.

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

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

• Modeling population growth and decay
• Calculating interest rates and investments
• Analyzing data and making predictions
• Solving problems in physics, engineering, and computer science

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

A: You can practice solving exponential equations and logarithms by working through problems in a textbook or online resource, or by using a calculator or computer program to generate random problems. You can also try solving problems on your own and then checking your solution with a calculator or computer program.

Q: What are some resources for learning more about exponential equations and logarithms?

A: Some resources for learning more about exponential equations and logarithms include:

• Textbooks and online resources such as Khan Academy and Mathway
• Calculators and computer programs such as Wolfram Alpha and Desmos
• Online communities and forums such as Reddit's r/learnmath and r/math
• Teachers and tutors who can provide one-on-one instruction and support.