Solving Logarithms: A Step-by-Step Guide

Hey everyone, let's dive into a cool math problem today! We're gonna figure out the value of ${}^2\log 27 \times {}^3\log 18$. Don't worry if it looks a bit intimidating at first; we'll break it down step by step, so you can totally nail it. This is a common type of problem you might find in your math class or even on a standardized test. The key here is to understand the properties of logarithms and how to apply them. We'll be using some clever tricks to simplify the expression and arrive at the final answer. So, grab your pencils, and let's get started!

First things first, remember that the expression ${}^a\log b means "the logarithm of b to the base a". In simpler terms, it's asking, "To what power must we raise a to get b?" It's like a mathematical puzzle, and we have to use our knowledge of exponents and logarithms to solve it. Keep in mind that understanding the fundamental properties of logarithms is crucial for tackling these kinds of problems. These properties will allow us to rewrite the logarithmic expressions in forms that are easier to work with. For example, knowing that log_a(b*c) = log_a(b) + log_a(c) can be super helpful. So, as we go through the steps, try to remember these rules! Understanding these rules and how they work together is the key to solving logarithmic equations effectively. This is where it starts to get fun and interesting! Let's get our hands dirty.

Okay, let's start with the first part of the expression: ${}^2\log 27$. This asks, "To what power must we raise 2 to get 27?" Well, 2 raised to any whole number power won't give us exactly 27. But we can rewrite 27 as a power of 3, because 27 is 3 cubed (3³ = 27). This is going to be useful to simplify. So, we can rewrite the expression using the change of base rule. The change of base rule lets us express a logarithm in terms of logarithms with a different base. It states that ${}^a\log b = ({}^c\log b)/({}^c\log a) for any valid base c. This gives us more flexibility in how we manipulate logarithmic expressions. Because, sometimes changing the base can make the simplification process much easier and cleaner.

Simplifying the First Logarithmic Term

Alright, guys, let's break down ${}^2\log 27 a bit more. We know that 27 can be written as 3³. So, we have ${}^2\log 3³. We can use a cool property of logarithms here: ${}^a\log b^c = c * {}^a\log b$. This means we can bring that exponent (3 in our case) down in front. Thus, ${}^2\log 27 = 3 * {}^2\log 3. Now this looks a lot simpler, right? It's often the case in math that rewriting things in different ways, or simplifying them by applying a rule, will make them easier to work with. Remember the rules of exponents and logarithms; they are your best friends in problems like these!

Now, let's move on to the second part of the original problem and focus on ${}^3\log 18. Again, we will rewrite the number 18 as a product of its prime factors. The prime factorization of 18 is 2 * 3². This is really helpful. Rewriting the number will allow us to then use the product rule of logarithms. This product rule states that the logarithm of a product is the sum of the logarithms. This is an essential logarithmic property that we can use to simplify our expression, and helps us to break down a more complex problem into smaller, more manageable parts. So, we can rewrite ${}^3\log 18 as ${}^3\log (2 * 3²). Remember that we can use the product rule of logarithms, which says that the log of a product is the sum of the logs. It's time to simplify this expression!

Solving the Second Logarithmic Term

Let's get cracking on ${}^3\log 18. As we said, 18 is the same as 2 * 3². This means we can rewrite ${}^3\log 18 as ${}^3\log (2 * 3²). Using the product rule of logarithms, we can further rewrite this. The product rule says that the log of a product is the sum of the logs. So, we get ${}^3\log 2 + {}^3\log 3².

Now, we can use another property of logarithms, that allows us to move an exponent in front. So, we can bring the exponent (the 2 in this case) down in front, which gives us ${}^3\log 2 + 2 * {}^3\log 3. And we're making some real progress here! We are getting closer to a final answer! We're using our tools, the properties of logarithms, to break down a seemingly complex problem into a bunch of smaller parts. Remember: understanding the fundamental principles makes these problems much more approachable.

Putting It All Together: The Grand Finale

Okay, so we have simplified both parts of our original expression. Now, let's bring it all together. Remember that the original problem was ${}^2\log 27 \times {}^3\log 18

We found that ${}^2\log 27 simplifies to 3 * {}^2\log 3. And we found that ${}^3\log 18 simplifies to ${}^3\log 2 + 2 * {}^3\log 3.

So, our expression becomes: (3 * {}^2\log 3) * ({}^3\log 2 + 2 * {}^3\log 3). Now, this might seem a little messy, but stay with me! This is where the magic happens. We can use the change of base rule. And remember the rule is ${}^a\log b = ({}^c\log b)/({}^c\log a). The cool thing about this rule is that we can change the base of the logarithm to anything we want, as long as it's the same base for both the numerator and the denominator. For example, if we have {}^2\log 3, we can rewrite it as ({}^3\log 3)/({}^3\log 2). That's awesome, right? Now let's put it into practice.

Applying the Change of Base Rule

Let's apply the change of base rule. We're going to rewrite {}^2\log 3 as ({}^3\log 3)/({}^3\log 2). We can substitute that into our current expression: (3 * ({}^3\log 3)/({}^3\log 2)) * ({}^3\log 2 + 2 * {}^3\log 3). This may look a bit daunting, but stick with it; we're close to the end. The key here is to simplify. Look for terms that can cancel each other out or be combined. Always remember the goal: to simplify the expression into something manageable. And always remember the rules and definitions of logarithms and exponents, because they are your tools. Math is a journey, and you are doing great.

Now, let's clean it up! Recall that {}^3\log 3 = 1 because the log of a number to the same base is always 1. So, 3 * ({}^3\log 3)/({}^3\log 2) simplifies to 3 / {}^3\log 2. This gives us (3 / {}^3\log 2) * ({}^3\log 2 + 2 * {}^3\log 3). Now we multiply this out, getting 3 + 6 * ({}^3\log 3)/({}^3\log 2). And then simplifying it more we get 3 + 6 * 1, which is just 3 + 6. Ta-da! Our final answer is 9. This means that the value of ${}^2\log 27 \times {}^3\log 18 is 9. High five! You did it!

Conclusion: You've Got This!

So, guys, we did it! We successfully solved ${}^2\log 27 \times {}^3\log 18, and the answer is 9. We broke down the problem step by step, used properties of logarithms like the change of base rule and the product rule. Always remember that, with practice, these types of problems become easier. Don't get discouraged if it seems tough at first; just keep practicing and reviewing the properties of logarithms. Each time you solve a problem like this, you're building a stronger foundation for your math skills. Keep up the awesome work, and keep exploring the wonderful world of mathematics! You've got this!