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

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 focus on solving the equation $3^{x-2}=3^7$, which is a classic example of an exponential equation. We will break down the solution step by step, using a combination of mathematical techniques and logical reasoning.

Understanding Exponential Equations

Exponential equations involve variables raised to a power, and the goal is to isolate the variable. In the equation $3^{x-2}=3^7$, we have a base of 3 raised to a power of $x-2$ on the left-hand side, and a base of 3 raised to a power of 7 on the right-hand side. The key to solving this equation is to recognize that the bases are the same, which allows us to set the exponents equal to each other.

Step 1: Setting the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other:

$$x-2=7$$

This is a simple linear equation, and we can solve for $x$ by adding 2 to both sides:

$$x=9$$

Step 2: Checking the Solution

To ensure that our solution is correct, we need to plug it back into the original equation and verify that it satisfies the equation. Substituting $x=9$ into the original equation, we get:

$$3^{9-2}=3^7$$

Simplifying the left-hand side, we get:

$$3^7=3^7$$

This confirms that our solution is correct.

Conclusion

Solving the equation $3^{x-2}=3^7$ requires a combination of mathematical techniques and logical reasoning. By setting the exponents equal to each other and solving for $x$, we can find the solution to the equation. In this case, the solution is $x=9$. This example illustrates the importance of understanding exponential equations and how to solve them using a variety of techniques.

Answer

The correct answer is:

• D. 9

Additional Examples

Here are a few more examples of exponential equations that you can try solving on your own:

• $2^{x+3}=2^5$
• $5^{x-1}=5^3$
• $4^{x+2}=4^6$

Remember to follow the same steps as before: set the exponents equal to each other, solve for $x$, and check the solution.

Tips and Tricks

Here are a few tips and tricks to help you solve exponential equations:

• Make sure the bases are the same before setting the exponents equal to each other.
• Use algebraic techniques, such as adding or subtracting the same value to both sides, to solve for $x$.
• Check the solution by plugging it back into the original equation.
• Practice, practice, practice! The more you practice solving exponential equations, the more comfortable you will become with the techniques and the more confident you will be in your ability to solve them.

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 typically written in the form $a^x=b$, where $a$ is the base, $x$ is the exponent, and $b$ is the result.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you need to isolate the variable by getting rid of the exponent. This can be done by using algebraic techniques, such as adding or subtracting the same value to both sides, or by using logarithms.

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

A: An exponential equation involves a variable raised to a power, while a linear equation involves a variable multiplied by a coefficient. For example, $2^x$ is an exponential equation, while $2x$ is a linear equation.

Q: Can I use logarithms to solve exponential equations?

A: Yes, logarithms can be used to solve exponential equations. By taking the logarithm of both sides of the equation, you can get rid of the exponent and solve for the variable.

Q: What is the logarithmic form of an exponential equation?

A: The logarithmic form of an exponential equation is $x = \log_a(b)$, where $a$ is the base, $b$ is the result, and $x$ is the exponent.

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

A: To use logarithms to solve an exponential equation, you need to take the logarithm of both sides of the equation. This will give you an equation in the form $x = \log_a(b)$, which you can then solve for $x$.

Q: What is the difference between a natural logarithm and a common logarithm?

A: A natural logarithm is a logarithm with a base of $e$, while a common logarithm is a logarithm with a base of 10.

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

A: Yes, you can use a calculator to solve exponential equations. Most calculators have a logarithm function that you can use to solve exponential equations.

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

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

• Not checking the solution to make sure it satisfies the original equation
• Not using the correct algebraic techniques to solve for the variable
• Not using logarithms when necessary
• Not checking the domain of the logarithm function

Q: How can I practice solving exponential equations?

A: You can practice solving exponential equations by working through examples and exercises in a textbook or online resource. You can also try solving exponential equations on your own, using a calculator or a computer program to check your answers.

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

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

• Modeling population growth and decay
• Modeling chemical reactions and nuclear decay
• Modeling financial growth and decay
• Modeling the spread of diseases

Conclusion

Solving exponential equations requires a combination of mathematical techniques and logical reasoning. By following the steps outlined in this article, you can solve exponential equations and become more confident in your ability to solve them. Remember to practice regularly and to check your solutions to ensure that they are correct. With practice and patience, you will become a master of solving exponential equations!