UFRGS 2017: Solving Log_5(x) = 2 Simply

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UFRGS 2017: Solving log_5(x) = 2 Simply

Hey everyone! Today, we're diving into a fun little math problem that appeared on the UFRGS (Universidade Federal do Rio Grande do Sul) exam back in 2017. It involves logarithms, and don't worry, we'll break it down so it's super easy to understand. Our mission is to solve the equation log_5(x) = 2. Sounds like fun? Let's jump right in!

Understanding Logarithms

Before we even think about solving this equation, let’s make sure we all understand what a logarithm actually is. Essentially, a logarithm answers the question: "To what power must I raise this base to get this number?" In our case, we have log_5(x) = 2. This is read as "log base 5 of x equals 2." So, we're asking: "To what power must we raise 5 to get x?"

Think of it like this:

If we have log_b(a) = c, that means b^c = a. The base b raised to the power of c gives us a. This is the fundamental relationship between logarithms and exponents, and it's super important to keep in mind. It's the key to unlocking all sorts of logarithmic puzzles.

Why are logarithms useful, guys?

Logarithms are incredibly useful in various fields, including science, engineering, and finance. They allow us to simplify complex calculations, especially those involving very large or very small numbers. They're also used in computer science, particularly in analyzing algorithms and data structures. So, understanding logarithms is not just about acing math tests; it's about building a solid foundation for many other areas of study and work.

For example, in chemistry, the pH scale uses logarithms to measure the acidity or alkalinity of a solution. In seismology, the Richter scale uses logarithms to measure the magnitude of earthquakes. The applications are truly endless! So, grasping this concept is more valuable than you might initially think.

Solving the Equation log_5(x) = 2

Now that we've refreshed our understanding of logarithms, let's get back to the problem at hand: solving log_5(x) = 2. Remember the relationship we just talked about? log_b(a) = c is the same as b^c = a. We can apply that directly to our equation.

In our case:

  • b = 5 (the base of the logarithm)
  • c = 2 (the value the logarithm equals)
  • a = x (the value we're trying to find)

So, log_5(x) = 2 translates to 5^2 = x. This is much easier to solve, right?

Let's calculate it:

5^2 means 5 multiplied by itself: 5 * 5 = 25. Therefore, x = 25. That's it! We've solved the equation. The value of x that satisfies the equation log_5(x) = 2 is 25. See? It wasn't as scary as it looked!

Let's Verify Our Solution:

It's always a good idea to check your answer to make sure you didn't make any mistakes. To verify that x = 25 is the correct solution, we can plug it back into the original equation:

log_5(25) = 2

Is this true? Well, we're asking: "To what power must we raise 5 to get 25?" And we know that 5^2 = 25, so yes, it is true! Our solution is correct.

Why This Type of Problem Matters

You might be wondering, “Okay, I can solve this problem now, but why does it even matter?” Well, problems like this one from the UFRGS exam test your understanding of fundamental mathematical concepts. They require you to not only memorize formulas but also understand how and why they work. This is crucial for developing problem-solving skills that can be applied to many different situations.

Real-World Applications:

As we mentioned earlier, logarithms have numerous real-world applications. Understanding them allows you to analyze and interpret data in fields like finance, engineering, and computer science. For example, in finance, logarithms are used to calculate compound interest and analyze investment growth. In engineering, they're used in signal processing and control systems. And in computer science, they're used to analyze the efficiency of algorithms.

Building a Strong Foundation:

Furthermore, mastering these types of problems builds a strong foundation for more advanced mathematical concepts. Logarithms are closely related to exponential functions, and understanding both is essential for calculus and other higher-level math courses. So, by tackling problems like this one, you're not just preparing for a test; you're preparing for future success in your academic and professional pursuits.

Common Mistakes to Avoid

When working with logarithms, it's easy to make mistakes if you're not careful. Here are a few common pitfalls to watch out for:

Forgetting the Base: Always remember the base of the logarithm. The base is crucial for converting between logarithmic and exponential forms. Forgetting the base can lead to incorrect calculations.

Incorrectly Applying the Definition: Make sure you understand the definition of a logarithm and how it relates to exponents. Confusing the base, the exponent, and the result can lead to errors.

Not Verifying the Solution: Always verify your solution by plugging it back into the original equation. This will help you catch any mistakes you might have made along the way.

Ignoring Domain Restrictions: Logarithms are only defined for positive arguments. If you end up with a negative value inside the logarithm, you know you've made a mistake.

By being aware of these common mistakes, you can avoid them and increase your chances of solving logarithmic equations correctly.

Practice Problems

Now that we've gone through the solution and discussed some common mistakes, let's test your understanding with a few practice problems:

  1. Solve for x: log_2(x) = 3
  2. Solve for x: log_3(x) = 4
  3. Solve for x: log_10(x) = 2

Try solving these problems on your own, and then check your answers. The solutions are provided below:

  1. x = 2^3 = 8
  2. x = 3^4 = 81
  3. x = 10^2 = 100

If you got all of these correct, congratulations! You have a solid understanding of how to solve logarithmic equations. If you struggled with any of them, don't worry. Just go back and review the concepts we discussed earlier, and keep practicing. The more you practice, the better you'll become!

Conclusion

So, there you have it, guys! We successfully solved the UFRGS 2017 math problem: log_5(x) = 2. Remember, the key is to understand the relationship between logarithms and exponents. Keep practicing, and you'll become a logarithm pro in no time! Understanding logarithms, as demonstrated by this UFRGS 2017 problem, is a valuable skill that extends far beyond the classroom. By mastering these fundamental concepts, you're equipping yourself with the tools to tackle a wide range of challenges in various fields. So keep practicing, stay curious, and never stop learning!