Solve For $r$:$5^r = \frac{1}{125}$

by ADMIN 40 views

=====================================================

Introduction

In this article, we will delve into solving exponential equations, specifically the equation $5^r = \frac{1}{125}$. This type of equation is a fundamental concept in mathematics, and understanding how to solve it is crucial for further studies in algebra, geometry, and other branches of mathematics. We will break down the solution step by step, making it easy to follow and understand.

Understanding Exponential Equations

Exponential equations are equations that involve an exponential expression, which is a number raised to a power. In this case, we have the equation $5^r = \frac{1}{125}$. The base of the exponential expression is 5, and the exponent is r. The equation states that 5 raised to the power of r is equal to 1/125.

Simplifying the Equation

To simplify the equation, we can rewrite 1/125 as a power of 5. Since 125 is equal to 5^3, we can rewrite 1/125 as 5^(-3). Therefore, the equation becomes:

5r=5−35^r = 5^{-3}

Using Properties of Exponents

Now that we have simplified the equation, we can use the properties of exponents to solve for r. One of the properties of exponents states that when two exponential expressions with the same base are equal, their exponents are also equal. In this case, we have:

5r=5−35^r = 5^{-3}

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

r=−3r = -3

Conclusion

In this article, we have solved the exponential equation $5^r = \frac{1}{125}$. We simplified the equation by rewriting 1/125 as a power of 5, and then used the properties of exponents to solve for r. The final answer is r = -3.

Real-World Applications

Exponential equations have many real-world applications, such as:

  • Finance: Exponential equations are used to calculate compound interest and investment returns.
  • Science: Exponential equations are used to model population growth, chemical reactions, and other natural phenomena.
  • Engineering: Exponential equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Tips and Tricks

Here are some tips and tricks for solving exponential equations:

  • Use the properties of exponents: When two exponential expressions with the same base are equal, their exponents are also equal.
  • Simplify the equation: Rewrite the equation in a simpler form, such as rewriting 1/125 as 5^(-3).
  • Use logarithms: If the equation is difficult to solve using the properties of exponents, try using logarithms to solve for the exponent.

Practice Problems

Here are some practice problems to help you practice solving exponential equations:

  • Problem 1: Solve the equation $2^r = 8$.
  • Problem 2: Solve the equation $3^r = 27$.
  • Problem 3: Solve the equation $4^r = 64$.

Solutions to Practice Problems

Here are the solutions to the practice problems:

  • Problem 1: $2^r = 8$

2r=232^r = 2^3

r=3r = 3

  • Problem 2: $3^r = 27$

3r=333^r = 3^3

r=3r = 3

  • Problem 3: $4^r = 64$

4r=434^r = 4^3

r=3r = 3

Conclusion

In this article, we have solved the exponential equation $5^r = \frac{1}{125}$. We simplified the equation by rewriting 1/125 as a power of 5, and then used the properties of exponents to solve for r. The final answer is r = -3. We also discussed the real-world applications of exponential equations and provided tips and tricks for solving them. Finally, we provided practice problems and solutions to help you practice solving exponential equations.

=============================

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponential equations.

Q: What is an exponential equation?

A: An exponential equation is an equation that involves an exponential expression, which is a number raised to a power. For example, the equation $5^r = \frac{1}{125}$ is an exponential equation.

Q: How do I solve an exponential equation?

A: To solve an exponential equation, you can use the properties of exponents. One of the properties of exponents states that when two exponential expressions with the same base are equal, their exponents are also equal. For example, if we have the equation $5^r = 5^{-3}$, we can equate the exponents and solve for r.

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

A: An exponential equation is an equation that involves an exponential expression, while a linear equation is an equation that involves a linear expression. For example, the equation $2x + 3 = 5$ is a linear equation, while the equation $2^x = 8$ is an exponential equation.

Q: Can I use logarithms to solve an exponential equation?

A: Yes, you can use logarithms to solve an exponential equation. Logarithms are the inverse of exponents, and they can be used to solve for the exponent in an exponential equation. For example, if we have the equation $2^x = 8$, we can take the logarithm of both sides and solve for x.

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

A: Exponential equations have many real-world applications, such as:

  • Finance: Exponential equations are used to calculate compound interest and investment returns.
  • Science: Exponential equations are used to model population growth, chemical reactions, and other natural phenomena.
  • Engineering: Exponential equations are used to design and optimize systems, such as electronic circuits and mechanical systems.

Q: How do I simplify an exponential equation?

A: To simplify an exponential equation, you can rewrite the equation in a simpler form. For example, if we have the equation $1/125 = 5^{-3}$, we can rewrite 1/125 as 5^(-3) and simplify the equation.

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 properties of exponents: Make sure to use the properties of exponents to solve for the exponent in an exponential equation.
  • Not simplifying the equation: Make sure to simplify the equation before solving for the exponent.
  • Not checking the solution: Make sure to check the solution to an exponential equation to ensure that it is correct.

Additional Resources

If you are struggling with exponential equations, here are some additional resources that may be helpful:

  • Online tutorials: There are many online tutorials and videos that can help you learn how to solve exponential equations.
  • Practice problems: Practice problems can help you build your skills and confidence when it comes to solving exponential equations.
  • Math textbooks: Math textbooks can provide you with a comprehensive overview of exponential equations and how to solve them.

Conclusion

In this article, we have answered some of the most frequently asked questions about exponential equations. We have discussed the properties of exponents, how to simplify an exponential equation, and some common mistakes to avoid when solving exponential equations. We have also provided some additional resources that may be helpful if you are struggling with exponential equations.