Solving ³log9 + ³log18 + ³log2: A Step-by-Step Guide

Hey guys! Today, we're diving into a fun math problem involving logarithms. Specifically, we're going to tackle the expression ³log9 + ³log18 + ³log2. Don't worry if it looks intimidating at first; we'll break it down step by step so it's super easy to understand. Whether you're a student prepping for an exam or just a math enthusiast, this guide is for you. So, let’s get started and unlock the secrets of logarithms together! We'll cover the fundamental concepts, the properties of logarithms we'll use, and then walk through the solution. You'll see, it's not as scary as it looks!

Understanding Logarithms: The Basics

Before we jump into solving the problem, let's quickly recap what logarithms are all about. At its heart, a logarithm is just another way of expressing exponents. Think of it like this: if we have an equation like aˣ = y, we can rewrite it in logarithmic form as logₐy = x. Here, a is the base, x is the exponent, and y is the result of raising a to the power of x. The logarithm asks the question: "To what power must we raise the base a to get y?"

So, for example, if we have 2³ = 8, we can express this logarithmically as log₂8 = 3. This tells us that we need to raise 2 to the power of 3 to get 8. Make sense? Great! Let's dive a little deeper. The base of a logarithm is super important. When we write logₐy, the little a subscript tells us what the base is. If you see a logarithm written without a base, like log y, it's generally understood that the base is 10. This is called the common logarithm. Another important logarithm is the natural logarithm, written as ln y, which has a base of e (Euler's number, approximately 2.71828).

In our problem, we're dealing with logarithms with a base of 3, written as ³log. This means we're asking: "To what power must we raise 3 to get a certain number?" For example, ³log9 asks, "To what power must we raise 3 to get 9?" The answer, of course, is 2, because 3² = 9. Understanding this fundamental concept is crucial for tackling more complex logarithmic expressions. Now that we've got the basics down, let's move on to the properties of logarithms that will help us solve our problem. These properties are like the secret tools in our math toolkit, allowing us to simplify and manipulate logarithmic expressions with ease. Knowing them well will make solving problems like this one a breeze!

Key Properties of Logarithms

To solve ³log9 + ³log18 + ³log2, we'll use a couple of key properties of logarithms. These properties are like magic spells that make complex expressions much simpler. Let's break them down:

1. Product Rule: This is our most important tool for this problem. The product rule states that the logarithm of a product is equal to the sum of the logarithms. Mathematically, it looks like this: logₐ(xy) = logₐx + logₐy. In simpler terms, if you're taking the logarithm of two numbers multiplied together, you can split it into the sum of the logarithms of each number. This works in reverse too, which is how we'll use it. If you have the sum of logarithms with the same base, you can combine them into a single logarithm of the product. Cool, right?

2. Change of Base Rule (Not Directly Used Here, But Good to Know): While we won't use this one directly in this problem, it's a handy property to have in your back pocket. The change of base rule allows you to convert a logarithm from one base to another. It says: logₐb = log꜀b / log꜀a. This is particularly useful when you need to evaluate logarithms on a calculator that doesn't have a specific base button. For example, if you need to find log₅15 but your calculator only has a base-10 logarithm button, you can use this rule to convert it to log₁₀15 / log₁₀5.

3. Power Rule (Also Not Directly Used Here): Another useful property is the power rule, which states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number. Mathematically: logₐ(xⁿ) = n logₐx. This rule can be handy for simplifying expressions where you have exponents inside the logarithm.

For our problem, the product rule is the star of the show. It's the key to combining our three logarithms into one, which will make the calculation much easier. Think of it as our secret weapon! With these properties in mind, we're now fully equipped to solve the problem. We've got our tools, we know how to use them, so let's get to it!

Step-by-Step Solution

Okay, let's get down to business and solve ³log9 + ³log18 + ³log2. Remember, we're going to use the product rule of logarithms, which says logₐx + logₐy = logₐ(xy). This rule is our best friend for this problem.

Step 1: Combine the Logarithms

We have three logarithmic terms being added together, and they all have the same base (3). This is perfect! We can use the product rule to combine them into a single logarithm. We multiply the arguments (the numbers inside the logarithms) together:

³log9 + ³log18 + ³log2 = ³log(9 × 18 × 2)

See how we've transformed three separate logarithms into one? This is the power of the product rule in action. Now, let's simplify the expression inside the logarithm.

Step 2: Simplify the Argument

Let's multiply those numbers together: 9 × 18 × 2. You can do this in any order you like, but let's go step by step:

• 9 × 18 = 162
• 162 × 2 = 324

So, our expression now looks like this:

³log(324)

We've made a lot of progress! We've combined the logarithms and simplified the argument. Now, we just need to evaluate this single logarithm.

Step 3: Evaluate the Logarithm

Now we need to figure out what power we need to raise 3 to, to get 324. In other words, we're solving for x in the equation:

3ˣ = 324

This might seem tricky at first, but let's think about it. We know that 3 raised to some power will give us 324. To figure out what that power is, let's try expressing 324 as a power of 3. We can do this by breaking 324 down into its prime factors. Alternatively, we can try different powers of 3 until we find the right one.

Let's try some powers of 3:

• 3¹ = 3
• 3² = 9
• 3³ = 27
• 3⁴ = 81
• 3⁵ = 243
• 3⁶ = 729

Hmm, it seems we skipped over 324! But let's not give up. Instead of guessing and checking, let's try another approach. Can we express 324 as a product of powers of 3 and other numbers? We know that 324 is divisible by 9 (since 3 + 2 + 4 = 9, which is divisible by 9). So, let's divide 324 by 9:

• 324 ÷ 9 = 36

Okay, so 324 = 9 × 36. We know that 9 is 3², but what about 36? Well, 36 is 4 × 9, which means 36 = 2² × 3². Let's put it all together:

• 324 = 9 × 36
• 324 = 3² × (2² × 3²)
• 324 = 3² × 2² × 3²
• 324 = 3⁴ × 2²

Ah, we're getting closer! But wait, we need to express 324 as a power of 3, not as a product of powers of 3 and 2. Let's go back to our original equation:

3ˣ = 324

We know that 324 is 9 multiplied by 36 (324 = 9 * 36). We also know that 9 is 3², so let's write it like this:

3ˣ = 3² * 36

Now, can we express 36 in terms of 3? Unfortunately, no. 36 isn't a power of 3. We've hit a bit of a roadblock here. Let’s rethink our approach. Instead of trying to express 324 as a simple power of 3, let's try a different tactic. Maybe there was an error in our calculation, or maybe 324 doesn’t simplify to a neat power of 3. Let's double-check our original problem and our steps so far.

³log9 + ³log18 + ³log2 = ³log(9 × 18 × 2)

³log(324)

Everything looks correct so far. Let's go back to our prime factorization of 324. We found that 324 = 3⁴ × 2². This is where we went slightly astray. We need to express 324 as 3 to some power, but the presence of 2² makes it impossible to express 324 as a simple power of 3. Hmmm… this suggests that the answer might not be a whole number. Let's try using the properties of logarithms in reverse to see if we missed something.

We have ³log324. Let's break 324 down into its factors again, but this time, let's focus on factors that are powers of 3. We know 324 = 9 × 36, and 9 is 3². So, we can write:

³log324 = ³log(9 × 36)

Now, let’s use the product rule in reverse:

³log(9 × 36) = ³log9 + ³log36

We know that ³log9 = 2 because 3² = 9. So, we have:

2 + ³log36

Now we need to figure out ³log36. Can we express 36 as 3 to some power? No, we can't. But we can write 36 as 6², so:

³log36 = ³log(6²)

This doesn’t seem to be leading us to a simple solution either. It appears we need a different strategy. Let's go back to our prime factorization of 324, which was 3⁴ × 2². Let’s rewrite our logarithm using this factorization:

³log324 = ³log(3⁴ × 2²)

Using the product rule, we can split this into:

³log(3⁴ × 2²) = ³log3⁴ + ³log2²

Now, using the power rule, we can bring the exponents down:

³log3⁴ + ³log2² = 4 × ³log3 + 2 × ³log2

We know that ³log3 = 1 (because 3¹ = 3), so:

4 × ³log3 + 2 × ³log2 = 4 × 1 + 2 × ³log2

= 4 + 2 × ³log2

This is an exact answer, but it's not a single number. It involves ³log2, which is an irrational number. If we needed a decimal approximation, we could use a calculator to find ³log2, but for an exact answer, this is as simplified as we can get.

So, the final answer is:

4 + 2 × ³log2

Final Answer:

The simplified value of ³log9 + ³log18 + ³log2 is 4 + 2 × ³log2. While we initially aimed for a single numerical answer, the presence of the factor 2² in 324 meant we couldn't express the result as a simple integer. This highlights the importance of understanding logarithm properties and how they can help us simplify expressions, even if we don't always arrive at a whole number answer. Great job working through this problem with me! Logarithms can be tricky, but with practice and a solid understanding of the rules, you'll become a pro in no time.

Conclusion

So, guys, we made it! We successfully solved the logarithmic expression ³log9 + ³log18 + ³log2. We started by understanding the basics of logarithms and their properties, particularly the product rule. We combined the logarithmic terms, simplified the expression, and then tackled the evaluation. While we didn't get a simple whole number answer, we arrived at the simplified form of 4 + 2 × ³log2. This problem perfectly illustrates how important it is to understand the fundamental properties of logarithms. By using the product rule, we were able to transform a seemingly complex expression into something much more manageable. Remember, math isn't just about getting the right answer; it's about the journey of problem-solving and understanding the concepts along the way.

Keep practicing, keep exploring, and you'll find that logarithms, like many areas of math, become much clearer with time and effort. And remember, if you ever get stuck, don't hesitate to break the problem down into smaller steps, revisit the fundamental properties, and try different approaches. Happy calculating, and I'll see you in the next math adventure!