Calculating Log 75: A Step-by-Step Guide

Hey guys! Ever been stuck trying to figure out the logarithm of a number? It can seem a little tricky at first, but with a few simple steps, you can totally nail it. Today, we're going to break down how to calculate the value of log 75, given that we know log 3 = 0.4771 and log 2 = 0.3010. This is a classic math problem that shows how logarithms work and how we can use known values to find the unknown ones. So, let’s dive right in and make sure you understand how to solve this, alright? We’ll take it slow and easy, so you won’t miss a thing. The key here is to use the properties of logarithms to simplify the expression and then use the given values to get our answer.

Before we start, let's quickly recap what logarithms are all about. In simple terms, a logarithm is the inverse operation to exponentiation. When we see log x, it's usually the common logarithm, which has a base of 10. So, log x = y means 10^y = x. Got it? Now we can start by expressing 75 as a product of its prime factors. This is a super important first step. Breaking down the number into its prime factors makes it easier to use the given log values. We can express 75 as 3 * 25. Then, we can simplify this further because 25 is also a perfect square (5 * 5). That gives us 3 * 5 * 5, or 3 * 5^2. Remember this step, it's like the secret sauce to the whole problem. This step will enable us to use the values of log 2 and log 3 that were already given to us. We will rewrite the given expression in terms of the given logs.

Breaking Down Log 75

Alright, so our first goal is to express 75 in terms of the prime factors of the given logarithms. Let's start with the basics. As we said before, 75 can be factored into its prime factors. The prime factorization of 75 is 3 * 5 * 5, which can also be written as 3 * 5^2. Now that we have the prime factorization, we can start using the properties of logarithms. One of the fundamental properties of logarithms is that the logarithm of a product is the sum of the logarithms. This means that log (a * b) = log a + log b. We can use this property to break down log 75 into simpler terms. Using this property, we can rewrite log 75 as log (3 * 5^2), which further breaks down into log 3 + log (5^2). See how this is getting easier already?

Now, there’s another property of logarithms that comes in handy here: the logarithm of a power. This property states that log (a^b) = b * log a. We can use this property to simplify log (5^2). Applying this property, log (5^2) becomes 2 * log 5. So, our equation now looks like this: log 75 = log 3 + 2 * log 5. Great, right? We're making progress. Now, we're one step closer to solving this problem by using the given log values. This step allows us to get closer to the final answer. We already know the value of log 3 (0.4771), but we don't have the value of log 5 directly. So, we'll need to figure that out next, and we will get our answer.

The Trick to Find Log 5

Okay, guys, here’s where we get a bit clever. We need to find log 5, but we don't have it directly. However, we do know log 2 and we know a handy trick. The cool thing is that we can express 5 in terms of 10 and 2. How, you ask? Well, 5 is equal to 10/2. So we can write log 5 as log (10/2). And guess what? This is another awesome place to use the properties of logarithms. Because the logarithm of a quotient is the difference of the logarithms, so log (a/b) = log a - log b. Using this property, log (10/2) becomes log 10 - log 2. Now this is starting to look good, isn’t it? Remember, when we just write log, it's the base-10 logarithm. So, log 10 = 1 because 10^1 = 10. And we already know log 2 = 0.3010. We can plug these values into our equation: log 5 = 1 - 0.3010, which equals 0.6990. Now we've found the value of log 5. We’re almost there, I promise!

Now that we know log 5, we can put everything together. We had log 75 = log 3 + 2 * log 5. We know log 3 = 0.4771, and we've calculated log 5 = 0.6990. So, we can plug these values into the equation to get log 75 = 0.4771 + 2 * 0.6990. Let's do the math: 2 * 0.6990 = 1.3980. Now, add that to 0.4771: 0.4771 + 1.3980 = 1.8751. So, log 75 = 1.8751. And there you have it, folks! We've successfully calculated the value of log 75 using the given values and the properties of logarithms. Wasn't that fun? With a bit of practice, you’ll be solving these problems in no time. Always remember to break down the problem into smaller, manageable steps. Identify the prime factors, apply the logarithm properties, and use the known values.

Putting It All Together

Let’s recap the whole process, just to make sure everything clicks. First, we started with log 75. Then, we broke down 75 into its prime factors: 3 * 5 * 5 or 3 * 5^2. Using the properties of logarithms, we rewrote log 75 as log 3 + 2 * log 5. Since we knew log 3, we had to find log 5. We cleverly expressed 5 as 10/2, which allowed us to use log 2 and the fact that log 10 = 1. We found log 5 by calculating log 10 - log 2, which gave us 0.6990. Finally, we plugged the values of log 3 and log 5 back into our original equation and did the math. The key takeaways from this exercise are understanding the prime factorization, using the properties of logarithms, and making sure you can express the unknown logarithms in terms of the values provided. This whole process helps you become a master of logarithms and is super useful in all kinds of mathematical problems. Keep practicing, and you'll get the hang of it in no time. Understanding the properties of logarithms is crucial for tackling more complex math problems.

So, there you have it, the solution to finding log 75! It might seem like a lot, but by breaking it down step by step, it becomes manageable. Remember, math is all about understanding the concepts and applying them in the right way. Keep practicing and exploring, and you'll find that these kinds of problems become easier and easier. If you encounter any other logarithm problems, try to break them down into smaller steps, identify the properties that apply, and always look for ways to relate the unknown values to the known ones. Well, that’s all for today, guys. Keep up the awesome work, and keep learning!