Select The Correct Answer.Solve The Exponential Equation For $x$.$625=5^{(7x-3)}$A. $ X = 2 X=2 X = 2 [/tex] B. $x=-2$ C. $x=-1$ D. $ X = 1 X=1 X = 1 [/tex]

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific exponential equation, $625=5^{(7x-3)}$, and provide a step-by-step guide on how to arrive at the correct solution.

Understanding Exponential Equations

Exponential equations involve a variable raised to a power, and the goal is to solve for the variable. In this case, we have the equation $625=5^{(7x-3)}$, where $x$ is the variable we need to solve for. The base of the exponent is $5$, and the exponent is $7x-3$.

Step 1: Simplify the Equation

To simplify the equation, we can start by rewriting $625$ as a power of $5$. We know that $625=5^4$, so we can rewrite the equation as:

54=5(7x−3)5^4=5^{(7x-3)}

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

4=7x−34=7x-3

Step 3: Solve for x

To solve for $x$, we can add $3$ to both sides of the equation:

4+3=7x−3+34+3=7x-3+3

7=7x7=7x

Step 4: Divide by 7

Finally, we can divide both sides of the equation by $7$ to solve for $x$:

77=7x7\frac{7}{7}=\frac{7x}{7}

1=x1=x

Conclusion

Therefore, the correct solution to the exponential equation $625=5^{(7x-3)}$ is $x=1$. This is the only option that satisfies the equation, making it the correct answer.

Answer

The correct answer is:

  • D. $x=1$

Discussion

This problem requires a strong understanding of exponential equations and the ability to simplify and solve them. The key concept here is that when the bases are the same, we can equate the exponents, and then solve for the variable. This is a fundamental skill that is used in a wide range of mathematical applications, from algebra to calculus.

Tips and Tricks

  • When solving exponential equations, always start by simplifying the equation and rewriting the bases as powers of the same base.
  • Use the property of equality to equate the exponents when the bases are the same.
  • Finally, solve for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Practice Problems

If you want to practice solving exponential equations, try the following problems:

  • 82=8(3x−2)8^2=8^{(3x-2)}

  • 273=27(4x−1)27^3=27^{(4x-1)}

  • 162=16(5x−3)16^2=16^{(5x-3)}

Conclusion

Q: What is an exponential equation?

A: An exponential equation is a mathematical equation that involves a variable raised to a power. It is a type of equation that can be solved using the properties of exponents.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can start by rewriting the bases as powers of the same base. For example, if you have the equation $82=8{(3x-2)}$, you can rewrite it as $82=82$.

Q: What is the property of equality that I can use to solve exponential equations?

A: The property of equality that you can use to solve exponential equations is that when the bases are the same, you can equate the exponents. For example, if you have the equation $82=8{(3x-2)}$, you can equate the exponents to get $2=3x-2$.

Q: How do I solve for the variable in an exponential equation?

A: To solve for the variable in an exponential equation, you can use the property of equality to equate the exponents, and then solve for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

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

  • Not simplifying the equation before solving for the variable
  • Not equating the exponents when the bases are the same
  • Not solving for the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through example problems, such as the ones listed below:

  • 82=8(3x−2)8^2=8^{(3x-2)}

  • 273=27(4x−1)27^3=27^{(4x-1)}

  • 162=16(5x−3)16^2=16^{(5x-3)}

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

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

  • Modeling population growth
  • Calculating compound interest
  • Determining the half-life of a radioactive substance
  • Solving problems involving exponential decay or growth

Q: Can I use a calculator to solve exponential equations?

A: Yes, you can use a calculator to solve exponential equations. However, it's always a good idea to check your work by hand to make sure that the calculator is giving you the correct answer.

Q: How can I check my work when solving exponential equations?

A: You can check your work by plugging your solution back into the original equation and making sure that it is true. For example, if you solve the equation $82=8{(3x-2)}$ and get $x=1$, you can plug $x=1$ back into the original equation to make sure that it is true.

Conclusion

Solving exponential equations can be a challenging task, but with practice and patience, you can become proficient in solving them. Remember to simplify the equation, equate the exponents, and solve for the variable to arrive at the correct answer. If you have any more questions or need further clarification, don't hesitate to ask.