Solving Logarithmic Equations: Log₃(36) + Log₃(100) - Log₃(4)
Hey guys! Let's dive into this interesting math problem involving logarithms. We're going to figure out the value of log₃(36) + log₃(100) - log₃(4) - log₃(4). Sounds a bit intimidating, right? But don't worry, we'll break it down step by step and make it super easy to understand. Logarithms might seem tricky at first, but once you grasp the basic rules, they become quite manageable. So, grab your thinking caps, and let's get started!
Understanding the Problem
Before we jump into solving, let’s make sure we understand what the problem is asking. We have an expression with several logarithms that we need to simplify and evaluate. The key here is to remember the properties of logarithms, which will allow us to combine and simplify these terms. Our main goal is to use these properties to reduce the expression into a single logarithm, or even better, a simple number. Remember, understanding the question is half the battle! We need to manipulate the given logarithmic expression using the fundamental rules of logarithms. We have a combination of addition and subtraction of logarithmic terms, all with the same base (3). This is a good sign because it means we can use the properties of logarithms to simplify the expression. The properties we'll be focusing on are the product rule and the quotient rule. The product rule states that logₐ(x) + logₐ(y) = logₐ(xy), and the quotient rule states that logₐ(x) - logₐ(y) = logₐ(x/y). These rules are the bread and butter of simplifying logarithmic expressions. Recognizing that all logarithms in the expression have the same base (3) is crucial. This allows us to combine them using logarithmic properties. If the bases were different, we would need to use a change of base formula, which adds another layer of complexity. So, having the same base simplifies our work significantly. The expression involves both addition and subtraction of logarithms. This means we will be applying both the product and quotient rules. First, we'll combine the terms being added using the product rule, and then we'll deal with the subtraction using the quotient rule. It's like following a recipe – each rule has its place in the process.
Step-by-Step Solution
Okay, let’s get down to the nitty-gritty and solve this! We'll take it one step at a time, so you can follow along easily. We'll be using the properties of logarithms to combine the terms. Remember, logarithms are just exponents in disguise, so the rules might seem familiar if you've worked with exponents before.
1. Combine Addition Using the Product Rule
The first step is to use the product rule to combine the addition parts of the expression. Remember the product rule? It says that logₐ(x) + logₐ(y) = logₐ(xy). Applying this to our problem, we have:
log₃(36) + log₃(100) = log₃(36 * 100) = log₃(3600)
So, we've transformed the sum of two logarithms into a single logarithm. This is a great start! The product rule is your friend when you see logarithms being added together with the same base. It allows you to multiply the arguments inside the logarithm. This is a fundamental property, and it's essential for simplifying logarithmic expressions. Make sure you're comfortable with this rule before moving on.
2. Combine Subtraction Using the Quotient Rule
Now, let’s tackle the subtraction parts. We'll use the quotient rule, which states that logₐ(x) - logₐ(y) = logₐ(x/y). Our expression now looks like this:
log₃(3600) - log₃(4) - log₃(4)
We can apply the quotient rule twice, or we can combine the subtractions first. Let's combine the subtractions:
log₃(4) + log₃(4) = log₃(4 * 4) = log₃(16)
Now our expression is:
log₃(3600) - log₃(16)
Applying the quotient rule, we get:
log₃(3600 / 16) = log₃(225)
The quotient rule is the counterpart to the product rule. When you see logarithms being subtracted, the quotient rule allows you to divide the arguments inside the logarithms. Just like the product rule, this is a crucial tool in simplifying logarithmic expressions. Remember that the order matters in subtraction, so be sure to divide the correct numbers.
3. Simplify the Logarithm
We're getting closer! We now have log₃(225). Can we simplify this further? We need to find out if 225 can be expressed as a power of 3. Let’s think... Is there an integer exponent that we can raise 3 to, to get 225? We know that 3 raised to some power will give us the answer. We need to find that power. This is where recognizing perfect squares and powers comes in handy. If we can express 225 as a power of 3, the logarithm will simplify nicely. If not, we might need to use a calculator to find an approximate value, but let's hope we can find an exact solution!
Let's break down 225. We know that 225 is 15 squared (15 * 15 = 225). Now, can we express 15 using factors of 3? Yes! 15 is 3 * 5. So, 225 = 15 * 15 = (3 * 5) * (3 * 5) = 3² * 5². This doesn't directly give us a power of 3, but it's a step in the right direction. But wait! Can we express 225 as 3 to the power of something? Let's try some powers of 3:
- 3¹ = 3
- 3² = 9
- 3³ = 27
- 3⁴ = 81
- 3⁵ = 243
Oops! It seems like 225 is not a direct power of 3. However, we made a small error in our previous calculation. Let's revisit log₃(225). We should recognize that 225 is 15^2. And 15 is 3 * 5. So, we cannot directly express 225 as a power of 3. This means we need to rethink our approach slightly.
Let's go back to our simplified expression: log₃(225). We need to find the exponent to which we must raise 3 to get 225. We can rewrite 225 as 3^x. So, we have:
3^x = 225
However, as we determined earlier, 225 is not a direct power of 3. But we also know that 225 = 15 * 15 = (3 * 5) * (3 * 5) = 3² * 5². This doesn’t directly translate to a simple integer exponent for 3. This indicates we might have made a mistake in our earlier steps or that the problem is designed to have an integer solution. Let’s go back and check our work to ensure we haven’t missed anything.
Going back to our steps, we had:
log₃(36) + log₃(100) - log₃(4) - log₃(4)
log₃(36 * 100) - log₃(4 * 4)
log₃(3600) - log₃(16)
log₃(3600 / 16)
log₃(225)
Everything seems correct so far. So, let's try expressing 225 in terms of its prime factors: 225 = 3² * 5². Now, we can rewrite our logarithm as:
log₃(3² * 5²) = log₃(3²) + log₃(5²) = 2 * log₃(3) + log₃(25)
Since log₃(3) = 1, we have:
2 + log₃(25)
This doesn’t seem to simplify to a nice integer value. Let's backtrack again and check for any arithmetic errors.
After careful review, the steps are logically sound. The most likely explanation is that there might have been a mistake in the original problem statement or the intended answer. Let’s consider a hypothetical scenario where the last term was log₃(9) instead of log₃(4). In that case, the calculation would be:
log₃(36) + log₃(100) - log₃(4) - log₃(9)
log₃(3600) - log₃(36)
log₃(3600 / 36)
log₃(100)
This still doesn’t give us a clean integer answer. However, if the problem intended to have + log₃(4) instead of - log₃(4) at the end, it would change things significantly:
log₃(36) + log₃(100) - log₃(4) + log₃(4)
log₃(3600)
log₃(3600) = log₃(36 * 100) = log₃(6² * 10²) = log₃((6 * 10)²) = log₃(60²)
This also doesn't lead to a simple solution. Given our calculations and the properties of logarithms, it seems the original problem, as stated, does not result in a straightforward integer answer. Therefore, we might need to re-examine the initial problem statement or consider if an approximation is acceptable.
It seems there's no direct integer solution. Let's double-check the original question and our steps one more time to be absolutely sure.
Final Calculation and Answer
Okay, after triple-checking everything, it seems like our calculations are correct, and 225 cannot be simplified further as a power of 3. This means log₃(225) is the simplest form we can get to without using a calculator to find a decimal approximation.
However, since this is a multiple-choice question, and none of the options (1, 2, 3, 4, 5) seem to fit our answer, it's possible there might be a typo in the question or the options provided. In a real-world scenario, you might want to double-check with your teacher or the source of the problem.
But, let's try to approximate the value to see which option is closest. We know that:
- 3⁴ = 81
- 3⁵ = 243
Since 225 is between 81 and 243, log₃(225) will be between 4 and 5. It's closer to 243, so it should be a bit closer to 5. Among the given options, 4 would be the closest approximation.
Therefore, based on our calculations and the given options, the closest answer is D. 4, even though it's not an exact match.
Key Takeaways
So, what did we learn from this problem? Here are some key takeaways:
- Master the Logarithm Properties: The product rule and quotient rule are your best friends when simplifying logarithmic expressions. Make sure you know them inside and out!
- Step-by-Step Approach: Break down complex problems into smaller, manageable steps. This makes it easier to keep track of your work and avoid mistakes.
- Double-Check Your Work: Always, always, always double-check your calculations! It's easy to make a small mistake, and it can throw off your entire answer.
- Think About the Question: Make sure you understand what the question is asking before you start solving. This will help you choose the right approach.
- Approximation When Needed: If you can't find an exact answer, try to approximate the value. This can help you choose the closest option in multiple-choice questions.
Practice Makes Perfect
Logarithms can be tricky, but with practice, you'll become a pro in no time! Try solving more problems like this one, and don't be afraid to ask for help if you get stuck. Keep practicing, and you'll master logarithms in no time!
I hope this explanation helped you guys understand how to solve this logarithmic equation. Remember, math is like a puzzle, and it's all about finding the right pieces to fit together. Keep practicing, and you'll become a math whiz in no time!