Solve The Equation: $ 5^{x-3} = 25^{x-5} $

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Introduction

Exponential equations are a fundamental concept in mathematics, and solving them requires a deep understanding of the properties of exponents. In this article, we will focus on solving the equation $ 5^{x-3} = 25^{x-5} $, which involves manipulating exponential expressions and using algebraic techniques to isolate the variable.

Understanding Exponential Equations

Exponential equations involve variables in the exponent, and they can be written in the form $ a^x = b^y $, where $ a $ and $ b $ are constants, and $ x $ and $ y $ are variables. In the equation $ 5^{x-3} = 25^{x-5} $, we have two exponential expressions with the same base, but different exponents.

Manipulating Exponential Expressions

To solve the equation $ 5^{x-3} = 25^{x-5} $, we need to manipulate the exponential expressions to make them easier to work with. We can start by rewriting the equation using the fact that $ 25 = 5^2 $. This gives us:

5x−3=(52)x−5 5^{x-3} = (5^2)^{x-5}

Using Exponent Rules

Now that we have rewritten the equation, we can use exponent rules to simplify it further. Specifically, we can use the rule that $ (am)n = a^{mn} $. Applying this rule to the equation, we get:

5x−3=52(x−5) 5^{x-3} = 5^{2(x-5)}

Equating Exponents

Since the bases are the same, we can equate the exponents to solve for the variable. This gives us:

x−3=2(x−5) x - 3 = 2(x - 5)

Solving for the Variable

Now that we have a linear equation, we can solve for the variable using basic algebra. We can start by distributing the 2 on the right-hand side:

x−3=2x−10 x - 3 = 2x - 10

Isolating the Variable

Next, we can isolate the variable by subtracting $ x $ from both sides:

−3=x−10 -3 = x - 10

Simplifying the Equation

Finally, we can simplify the equation by adding 10 to both sides:

7=x 7 = x

Conclusion

In this article, we have solved the equation $ 5^{x-3} = 25^{x-5} $ using a combination of exponent rules and algebraic techniques. We started by rewriting the equation using the fact that $ 25 = 5^2 $, and then used exponent rules to simplify it further. Finally, we equated the exponents and solved for the variable using basic algebra. The solution to the equation is $ x = 7 $.

Additional Tips and Tricks

  • When solving exponential equations, it's often helpful to rewrite the equation using the fact that $ a^m = b^n $ implies $ a = b^{n/m} $.
  • Exponent rules can be used to simplify exponential expressions and make them easier to work with.
  • Equating exponents is a powerful technique for solving exponential equations, but it requires careful attention to the bases and exponents.

Common Mistakes to Avoid

  • When solving exponential equations, it's easy to get caught up in the algebra and forget to check the bases and exponents.
  • Make sure to use exponent rules carefully and consistently, and avoid making mistakes that can lead to incorrect solutions.
  • Equating exponents requires careful attention to the bases and exponents, so make sure to double-check your work.

Real-World Applications

Exponential equations have many real-world applications, including:

  • Modeling population growth and decay
  • Analyzing financial data and predicting stock prices
  • Understanding chemical reactions and kinetics
  • Solving problems in physics and engineering

Final Thoughts

Solving exponential equations requires a deep understanding of the properties of exponents and a combination of algebraic techniques. By following the steps outlined in this article, you can solve equations like $ 5^{x-3} = 25^{x-5} $ and develop a deeper understanding of exponential equations. Remember to use exponent rules carefully, equate exponents with caution, and double-check your work to avoid common mistakes.

Introduction

In our previous article, we explored the steps to solve the equation $ 5^{x-3} = 25^{x-5} $ using a combination of exponent rules and algebraic techniques. However, we know that practice makes perfect, and the best way to learn is by doing. In this article, we will provide a Q&A guide to help you practice solving exponential equations and build your confidence in tackling more complex problems.

Q1: What is the first step in solving an exponential equation?

A1: The first step in solving an exponential equation is to rewrite the equation using the fact that the bases are the same. This will allow you to use exponent rules to simplify the equation and make it easier to work with.

Q2: How do I rewrite an exponential equation using the same base?

A2: To rewrite an exponential equation using the same base, you can use the fact that $ a^m = b^n $ implies $ a = b^{n/m} $. For example, if you have the equation $ 5^{x-3} = 25^{x-5} $, you can rewrite it as $ 5^{x-3} = (52){x-5} $.

Q3: What is the next step after rewriting the equation?

A3: After rewriting the equation, the next step is to use exponent rules to simplify it further. This may involve using the rule that $ (am)n = a^{mn} $ or other exponent rules to make the equation easier to work with.

Q4: How do I equate the exponents in an exponential equation?

A4: To equate the exponents in an exponential equation, you need to make sure that the bases are the same. Once you have the same base, you can equate the exponents by setting the exponents equal to each other. For example, if you have the equation $ 5^{x-3} = 5^{2(x-5)} $, you can equate the exponents by setting $ x - 3 = 2(x - 5) $.

Q5: What is the final step in solving an exponential equation?

A5: The final step in solving an exponential equation is to solve for the variable using basic algebra. This may involve isolating the variable, simplifying the equation, and solving for the variable.

Q6: What are some common mistakes to avoid when solving exponential equations?

A6: Some common mistakes to avoid when solving exponential equations include:

  • Forgetting to check the bases and exponents
  • Making mistakes when using exponent rules
  • Failing to equate the exponents correctly
  • Not double-checking your work

Q7: What are some real-world applications of exponential equations?

A7: Exponential equations have many real-world applications, including:

  • Modeling population growth and decay
  • Analyzing financial data and predicting stock prices
  • Understanding chemical reactions and kinetics
  • Solving problems in physics and engineering

Q8: How can I practice solving exponential equations?

A8: You can practice solving exponential equations by working through examples and exercises, such as the one we solved in our previous article. You can also try creating your own problems and solving them using the steps outlined in this article.

Q9: What are some additional tips and tricks for solving exponential equations?

A9: Some additional tips and tricks for solving exponential equations include:

  • Using exponent rules carefully and consistently
  • Equating exponents with caution
  • Double-checking your work
  • Practicing regularly to build your confidence and skills

Q10: Where can I find more resources and practice problems for solving exponential equations?

A10: You can find more resources and practice problems for solving exponential equations by searching online, checking out math textbooks and workbooks, or seeking help from a tutor or teacher.

Conclusion

Solving exponential equations requires a combination of exponent rules and algebraic techniques. By following the steps outlined in this article and practicing regularly, you can build your confidence and skills in solving exponential equations. Remember to use exponent rules carefully, equate exponents with caution, and double-check your work to avoid common mistakes.