Solve $3^{x-2} = 3^7$.A. 2 B. 5 C. 7 D. 9

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Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. In this equation, the base is the same on both sides, and we need to solve for the exponent. The equation $3^{x-2} = 3^7$ is a classic example of an exponential equation, and in this article, we will learn how to solve it.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential function, which is a function that raises a number to a power. In this case, the base is 3, and the exponent is x-2. The equation $3^{x-2} = 3^7$ can be rewritten as $3^{x-2} = 3^7$, where the base is the same on both sides.

Solving Exponential Equations

To solve an exponential equation, we need to set the exponents equal to each other. In this case, we can set x-2 equal to 7, since the bases are the same. This gives us the equation x-2 = 7.

Simplifying the Equation

To solve for x, we need to isolate x on one side of the equation. We can do this by adding 2 to both sides of the equation. This gives us x = 7 + 2.

Evaluating the Solution

Now that we have the solution x = 9, we can evaluate it to make sure it is correct. We can plug x = 9 back into the original equation $3^{x-2} = 3^7$ to see if it is true.

Checking the Solution

To check the solution, we can plug x = 9 back into the original equation. This gives us $3^{9-2} = 3^7$, which simplifies to $3^7 = 3^7$. This shows that the solution x = 9 is correct.

Conclusion

In this article, we learned how to solve the exponential equation $3^{x-2} = 3^7$. We started by understanding the equation and setting the exponents equal to each other. We then simplified the equation and evaluated the solution to make sure it was correct. The final answer is x = 9.

Frequently Asked Questions

  • Q: What is the base of the exponential equation? A: The base of the exponential equation is 3.
  • Q: What is the exponent of the exponential equation? A: The exponent of the exponential equation is x-2.
  • Q: How do we solve an exponential equation? A: To solve an exponential equation, we need to set the exponents equal to each other.
  • Q: What is the solution to the exponential equation? A: The solution to the exponential equation is x = 9.

Step-by-Step Solution

  1. Understand the equation and set the exponents equal to each other.
  2. Simplify the equation by adding 2 to both sides.
  3. Evaluate the solution by plugging x = 9 back into the original equation.
  4. Check the solution by simplifying the equation.

Key Takeaways

  • Exponential equations involve an exponential function.
  • To solve an exponential equation, we need to set the exponents equal to each other.
  • The base of the exponential equation is 3.
  • The exponent of the exponential equation is x-2.
  • The solution to the exponential equation is x = 9.

Final Answer

The final answer is x = 9.

Introduction

Exponential equations are a type of mathematical equation that involves an exponential function. In this article, we will answer some frequently asked questions about exponential equations.

Q: What is an exponential equation?

A: An exponential equation is a type of mathematical equation that involves an exponential function. It is an equation that has a base raised to a power, and the base is the same on both sides of the equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to set the exponents equal to each other. This is because the bases are the same on both sides of the equation, so the exponents must be equal.

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 $3^{x-2} = 3^7$, the base is 3.

Q: What is the exponent of an exponential equation?

A: The exponent of an exponential equation is the power to which the base is raised. In the equation $3^{x-2} = 3^7$, the exponent is x-2.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you need to use the properties of exponents. For example, if you have the equation $3^{x-2} = 3^7$, you can simplify it by adding 2 to both sides, which gives you x = 7 + 2.

Q: How do I evaluate the solution to an exponential equation?

A: To evaluate the solution to an exponential equation, you need to plug the solution back into the original equation. For example, if you have the solution x = 9, you can plug it back into the equation $3^{x-2} = 3^7$ to see if it is true.

Q: What is the solution to the exponential equation $3^{x-2} = 3^7$?

A: The solution to the exponential equation $3^{x-2} = 3^7$ is x = 9.

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

A: To check the solution to an exponential equation, you need to plug the solution back into the original equation and simplify it. For example, if you have the solution x = 9, you can plug it back into the equation $3^{x-2} = 3^7$ and simplify it to see if it is true.

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

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

  • Not setting the exponents equal to each other
  • Not simplifying the equation
  • Not evaluating the solution
  • Not checking the solution

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

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

  • Modeling population growth
  • Modeling chemical reactions
  • Modeling financial growth
  • Modeling physical phenomena

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

A: There are many software programs and online tools that can be used to solve exponential equations, including:

  • Graphing calculators
  • Online equation solvers
  • Computer algebra systems

Q: What are some tips for solving exponential equations?

A: Some tips for solving exponential equations include:

  • Read the problem carefully
  • Identify the base and exponent
  • Set the exponents equal to each other
  • Simplify the equation
  • Evaluate the solution
  • Check the solution

Conclusion

Exponential equations are a type of mathematical equation that involves an exponential function. In this article, we have answered some frequently asked questions about exponential equations, including how to solve them, how to simplify them, and how to evaluate the solution. We have also discussed some real-world applications of exponential equations and some tips for solving them.