UFRGS 2017: Solving Log_5(x) = 2 Simply

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UFRGS 2017: Solving log_5(x) = 2 Simply

Hey guys! Today, let's break down a classic logarithm problem from the UFRGS 2017 exam. We're tackling the equation log_5(x) = 2. Don't worry, it's simpler than it looks! We'll go through the steps nice and slow, so you can nail it every time. Understanding logarithms is super important for math and science, and this is a fundamental problem. Let's get started!

Understanding the Problem

First, let's make sure we understand exactly what the problem is asking. The equation log_5(x) = 2 is a logarithmic equation where we need to find the value of 'x'. In simpler terms, we need to find the number 'x' such that when 5 is raised to the power of 2, it equals 'x'. Logarithms are essentially the inverse operation of exponentiation. So, understanding this relationship is key to solving the problem. When you see a logarithm, think, "What power do I need to raise the base to, to get this number?" Here, the base is 5, and we want to know what power of 5 gives us 'x'. The equation tells us that 5 raised to the power of 2 equals 'x'. This is the core concept. Before diving into the solution, it's crucial to grasp this fundamental principle of logarithms. Remember, a logarithm isolates the exponent. This will help you convert it into exponential form, making it easier to solve for 'x'. So, always keep this in mind: logarithms are all about finding the exponent.

Converting Logarithmic Form to Exponential Form

Okay, so we have log_5(x) = 2. The golden rule here is to convert this logarithmic form into its equivalent exponential form. Remember, the logarithm asks: "What power do I need to raise the base (5 in this case) to, to get 'x'?" The equation tells us that the answer to that question is 2. So, in exponential form, this translates to 5^2 = x. See how we took the base of the logarithm (5), raised it to the power on the other side of the equation (2), and set that equal to the argument of the logarithm (x)? This conversion is super important and makes the problem much easier to solve. It's like translating from one language to another – once you know the translation, everything becomes clear. Make sure you practice this conversion with different logarithmic equations. For example, if you have log_3(9) = y, it converts to 3^y = 9. The more you practice, the more natural this conversion will become. This step is the foundation for solving logarithmic equations, and mastering it will help you tackle more complex problems later on. Remember, practice makes perfect! Don't skip this step; it's the key to unlocking the solution.

Solving for x

Now that we've converted our equation into exponential form, 5^2 = x, solving for 'x' is a breeze! All we need to do is calculate 5 squared. Remember, 5 squared means 5 multiplied by itself: 5 * 5. And what's 5 * 5? It's 25! So, x = 25. That's it! We've found the value of 'x' that satisfies the original equation. To double-check our answer, we can plug it back into the original logarithmic equation: log_5(25) = 2. This asks, "What power do I need to raise 5 to, to get 25?" And the answer is indeed 2, because 5^2 = 25. So, our solution is correct! Solving for 'x' in this case was straightforward because once we converted to exponential form, it was just a simple calculation. However, in more complex problems, you might need to do some algebraic manipulation after converting to exponential form. The key is to isolate 'x' on one side of the equation. But in this case, we got lucky, and the solution was right there after the conversion. So, always remember to double-check your answer by plugging it back into the original equation. This will help you avoid careless mistakes and ensure that your solution is correct.

The Answer

So, after converting the logarithmic equation log_5(x) = 2 into its exponential form, 5^2 = x, and performing the simple calculation, we found that x = 25. This means that 5 raised to the power of 2 equals 25, which satisfies the original equation. Therefore, the solution to the UFRGS 2017 problem is x = 25. This is a fundamental example of how to solve logarithmic equations, and it highlights the importance of understanding the relationship between logarithms and exponents. Remember, logarithms and exponents are just two sides of the same coin. Mastering the conversion between these two forms is essential for solving a wide range of mathematical problems. Now, you can confidently tackle similar logarithmic equations! You've learned how to convert from logarithmic to exponential form, how to solve for the unknown variable, and how to verify your answer. Great job! Keep practicing, and you'll become a logarithm pro in no time!

Key Takeaways

Alright, let's recap the main points so this sticks with you. First, always understand the relationship between logarithms and exponents. They're inverses of each other. Second, master the conversion from logarithmic form to exponential form. This is the key to simplifying the problem. In our case, log_5(x) = 2 became 5^2 = x. Third, solve for 'x' after the conversion. This might involve a simple calculation, like in our example (5^2 = 25), or it might require some algebraic manipulation. Fourth, always double-check your answer by plugging it back into the original equation. This ensures you didn't make any mistakes along the way. Finally, practice makes perfect! The more you practice solving logarithmic equations, the more comfortable and confident you'll become. Start with simple problems like this one, and gradually work your way up to more complex ones. And remember, don't be afraid to ask for help if you get stuck. There are plenty of resources available online and in textbooks. Keep learning, keep practicing, and you'll master logarithms in no time!

Practice Problems

Want to test your understanding? Here are a few practice problems similar to the UFRGS 2017 question. Try solving them on your own, and then check your answers with the solutions provided below.

  1. log_2(x) = 3
  2. log_3(x) = 2
  3. log_10(x) = 1
  4. log_4(x) = 2
  5. log_5(x) = 3

Solutions:

  1. x = 2^3 = 8
  2. x = 3^2 = 9
  3. x = 10^1 = 10
  4. x = 4^2 = 16
  5. x = 5^3 = 125

How did you do? Hopefully, you got them all right! If not, don't worry. Just go back and review the steps we discussed earlier. Remember to focus on understanding the relationship between logarithms and exponents and mastering the conversion between the two forms. And keep practicing! The more you practice, the better you'll become at solving these types of problems. These practice problems are designed to reinforce your understanding of the basic concepts. As you become more comfortable with these, you can start tackling more challenging problems that involve more complex algebraic manipulations. But always remember to start with the fundamentals and build from there. Good luck, and happy solving!