UFRGS 2017: Solve Log_5(x) = 2 Simply!
Hey everyone! Today, let's break down a cool math problem from the UFRGS 2017 exam. We're tackling the equation log5(x) = 2. Don't worry, it's not as scary as it looks! We'll go through it step-by-step so you can ace similar questions in the future. Whether you're a student prepping for exams or just a math enthusiast, this guide will help you understand the fundamentals of logarithms and how to solve them effortlessly. So, grab your thinking caps, and let's dive right in!
Understanding Logarithms
Before we jump into solving the equation, let's quickly recap what logarithms are all about. Logarithms are essentially the inverse operation to exponentiation. Think of it like this: if you have an equation like ab = c, the logarithm answers the question, "What power (b) do I need to raise 'a' to, in order to get 'c'?" This is written as loga(c) = b.
The 'a' in loga(c) is called the base of the logarithm. It's the number that's being raised to a power. The 'c' is the argument of the logarithm – the number you're trying to get to. And 'b' is the exponent – the power you need to raise 'a' to.
For example, log2(8) = 3 because 23 = 8. Simple, right? Understanding this relationship between logarithms and exponents is crucial for solving logarithmic equations. It allows us to switch between logarithmic form and exponential form, which is often the key to unlocking the solution. Remember, the logarithm is just asking, "What power do I need?"
Logarithms are used everywhere, from calculating the pH of a solution in chemistry to determining the magnitude of an earthquake on the Richter scale. They're also essential in computer science, particularly in analyzing algorithms and data structures. So, grasping the basics of logarithms is not just helpful for exams, but also for many real-world applications. Keep practicing, and you'll become a logarithm pro in no time!
Breaking Down the Equation: log₅(x) = 2
Okay, now that we've refreshed our understanding of logarithms, let's get back to the problem at hand: log5(x) = 2. This equation is telling us that the logarithm of 'x' to the base 5 is equal to 2. In other words, we need to find the value of 'x' that satisfies this condition.
To solve this, we need to convert the logarithmic equation into its equivalent exponential form. Remember, the relationship between logarithms and exponents is key here. The equation loga(c) = b can be rewritten as ab = c. Applying this to our problem, we have:
log5(x) = 2 --> 52 = x
See how we simply rewrote the equation? The base of the logarithm (5) becomes the base of the exponent, the result of the logarithm (2) becomes the exponent, and the argument of the logarithm (x) becomes the result of the exponentiation. Now, the equation is much easier to solve!
This conversion is the most important step in solving logarithmic equations. It transforms the problem from a logarithmic form, which can be tricky to work with, into an exponential form, which is usually straightforward to solve. Make sure you understand this conversion process inside and out. Practice converting different logarithmic equations into exponential form, and vice versa. The more you practice, the more comfortable you'll become with this transformation, and the easier it will be to solve logarithmic problems.
Solving for x
Alright, we've transformed the equation log5(x) = 2 into its exponential form: 52 = x. Now comes the easy part – solving for 'x'! All we need to do is calculate 5 squared, which is 5 multiplied by itself.
52 = 5 * 5 = 25
Therefore, 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 equation: log5(25) = 2. Is this true? Well, 5 raised to the power of 2 does indeed equal 25, so our answer is correct!
This step highlights the importance of understanding exponents. Knowing how to calculate exponents quickly and accurately is crucial for solving many math problems, including logarithmic equations. Make sure you're comfortable with basic exponents and how they work. Practice calculating different exponents, and you'll become much faster at solving equations like this.
Also, remember to always double-check your answer by plugging it back into the original equation. This is a great way to catch any mistakes you might have made along the way. It gives you confidence that your solution is correct and helps you avoid losing points on exams.
Verification
To be absolutely sure of our solution, let's verify it. We found that x = 25. So, we need to check if log5(25) really equals 2.
Think about it: What power do we need to raise 5 to, in order to get 25? Well, 52 = 25. So, the logarithm of 25 to the base 5 is indeed 2. This confirms that our solution is correct.
log5(25) = log5(52) = 2
Verification is a crucial step in problem-solving. It's not just about finding an answer; it's about making sure that your answer is correct. By verifying your solution, you can catch any mistakes you might have made and avoid losing points on exams. It also helps you build confidence in your problem-solving abilities.
There are several ways to verify your solution, depending on the type of problem you're solving. In this case, we simply plugged our solution back into the original equation and checked if it held true. For other types of problems, you might need to use different techniques, such as graphing the equation or using a different method to solve the problem and comparing the results. The key is to always take the time to verify your solution, no matter how confident you are in your answer.
Conclusion
So, there you have it! We've successfully solved the UFRGS 2017 math problem: log5(x) = 2. The solution is x = 25. Remember, the key to solving logarithmic equations is to convert them into their equivalent exponential form. Once you've done that, the rest is usually straightforward.
Solving logarithmic equations might seem daunting at first, but with practice, it becomes much easier. The more you practice, the more comfortable you'll become with logarithms and exponents, and the faster you'll be able to solve these types of problems. Remember to always review the fundamental concepts, such as the relationship between logarithms and exponents, and to practice converting between logarithmic and exponential forms.
Keep practicing, and you'll become a math whiz in no time! And remember, math is not just about getting the right answer; it's about understanding the process and developing your problem-solving skills. So, keep exploring, keep learning, and keep having fun with math!