Solving 3 Log54 + 3 Log18 - 3 Log12 A Comprehensive Guide

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Hey guys! Today, we're diving into a super interesting math problem that might seem a bit intimidating at first glance. We're going to break down how to solve the expression 3 Log54 + 3 Log18 - 3 Log12. Don’t worry, it’s not as scary as it looks! We’ll go through each step in detail, making sure you understand the logic behind it all. So, grab your pencils, and let’s get started!

Understanding Logarithms: The Basics

Before we jump into the problem, let’s quickly recap what logarithms are. In simple terms, a logarithm answers the question: “To what power must we raise a base number to get a specific value?” For example, the logarithm of 100 to the base 10 (written as log₁₀(100)) is 2 because 10² = 100. Understanding this basic concept is crucial for solving logarithmic expressions. Now, let’s translate this into our problem. We have 3 Log54 + 3 Log18 - 3 Log12. Notice that each term has a coefficient of 3, and we are dealing with common logarithms (base 10). The key here is to remember the properties of logarithms, which will help us simplify this expression.

Key Logarithmic Properties

There are a few properties of logarithms that we’ll use extensively in solving this problem. These include:

  1. Product Rule: logₐ(mn) = logₐ(m) + logₐ(n)
  2. Quotient Rule: logₐ(m/n) = logₐ(m) - logₐ(n)
  3. Power Rule: logₐ(mᵖ) = p * logₐ(m)

These rules are the building blocks for simplifying logarithmic expressions. Think of them as your secret weapons in tackling log problems. Let's see how we can apply these rules to our specific problem. First, we'll use the power rule to deal with the coefficient 3 in front of each logarithm. This will make the expression much easier to handle and set us up for the next steps in simplification. Remember, the goal is to break down the complex expression into smaller, manageable parts. By understanding and applying these properties, we can transform what seems like a difficult problem into a series of simpler steps.

Step-by-Step Solution

Now, let’s tackle the problem step by step. Remember our expression: 3 Log54 + 3 Log18 - 3 Log12. Our first move will be to use the power rule to bring the coefficient 3 inside each logarithm. This means we’ll rewrite the expression as:

Log(54³) + Log(18³) - Log(12³)

This step is crucial because it allows us to combine the logarithms using the product and quotient rules. Think of it as organizing your tools before starting a big project. By bringing the coefficients inside, we’ve set the stage for the next phase of simplification. Now, let's calculate the cubes of 54, 18, and 12. This might seem like a daunting task, but it’s a necessary step to move forward. So, grab your calculators, and let’s crunch these numbers!

Calculating the Cubes

Let's calculate the cubes of 54, 18, and 12:

  • 54³ = 54 * 54 * 54 = 157464
  • 18³ = 18 * 18 * 18 = 5832
  • 12³ = 12 * 12 * 12 = 1728

Now we can rewrite our expression as:

Log(157464) + Log(5832) - Log(1728)

This looks a lot cleaner already, doesn’t it? We've transformed the original expression into something much more manageable. The next step involves using the product and quotient rules to combine these logarithms. This is where the magic happens – we’ll condense multiple log terms into a single, simplified logarithm. It's like combining ingredients in a recipe to create the final dish. By applying these rules, we're one step closer to the final answer.

Applying the Product and Quotient Rules

Now, let’s use the product and quotient rules to simplify further. First, we’ll combine the addition of logarithms using the product rule:

Log(157464) + Log(5832) = Log(157464 * 5832)

Multiply 157464 by 5832:

157464 * 5832 = 918309648

So our expression becomes:

Log(918309648) - Log(1728)

Next, we’ll use the quotient rule to combine the subtraction of logarithms:

Log(918309648) - Log(1728) = Log(918309648 / 1728)

Now, divide 918309648 by 1728:

918309648 / 1728 = 531441

Our expression is now:

Log(531441)

We’ve come a long way, haven’t we? From the initial expression with multiple logarithms and coefficients, we’ve distilled it down to a single logarithm. This is a significant achievement! Now, the final step is to evaluate this logarithm. This involves figuring out what power of 10 gives us 531441. This might seem daunting, but there’s a clever trick to it, which we’ll explore next.

Evaluating the Final Logarithm

We’ve simplified our expression to Log(531441). Now, we need to evaluate this logarithm. To do this, we recognize that 531441 is a power of 3. Specifically, 531441 = 3¹². However, we need to express it in terms of powers of 10 since we’re dealing with a common logarithm (base 10). Let’s think about the number 531441 in a different way. Can we express it as 10 raised to some power? Well, not directly as an integer power. But we can recognize a pattern. Notice that:

531441 = 3¹² = (3²)⁶ = 9⁶

This is a helpful observation because 9 is close to 10, which is our base. Now, let’s rewrite the expression slightly to make it clearer:

Log(531441) = Log(9⁶)

Using the power rule again, we can bring the exponent outside the logarithm:

Log(9⁶) = 6 * Log(9)

Now we're dealing with Log(9), which is much easier to handle. We know that 9 is slightly less than 10, so Log(9) will be slightly less than Log(10), which is 1. To get a more precise value, we can use a calculator or logarithm table. The value of Log(9) is approximately 0.9542. So, we have:

6 * Log(9) ≈ 6 * 0.9542

Multiplying this out gives us:

6 * 0.9542 ≈ 5.7252

Therefore, the final answer is approximately 5.7252. We've successfully navigated through the problem, using the properties of logarithms to simplify and evaluate the expression. This is a fantastic accomplishment! We started with a seemingly complex problem and broke it down into manageable steps. Now, let’s summarize the entire process to reinforce what we’ve learned.

Summary of Steps

Let’s recap the steps we took to solve the problem 3 Log54 + 3 Log18 - 3 Log12:

  1. Apply the Power Rule: Rewrite the expression as Log(54³) + Log(18³) - Log(12³).
  2. Calculate the Cubes: Find the values of 54³, 18³, and 12³.
  3. Apply the Product Rule: Combine the addition of logarithms by multiplying the numbers inside the logarithms.
  4. Apply the Quotient Rule: Combine the subtraction of logarithms by dividing the numbers inside the logarithms.
  5. Evaluate the Final Logarithm: Recognize the result as a power of 3 and use logarithm properties to simplify and find the final answer.

By following these steps, we transformed a complex logarithmic expression into a simple value. This demonstrates the power of understanding and applying the fundamental properties of logarithms. Remember, math problems often look daunting at first, but by breaking them down into smaller steps and using the right tools, you can solve anything. Now, let’s discuss some common mistakes to avoid when solving similar problems.

Common Mistakes to Avoid

When solving logarithmic expressions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you get the correct answer. Let’s go through some of the most frequent errors:

  1. Incorrect Application of Logarithmic Properties: One of the most common mistakes is misapplying the product, quotient, or power rules. For example, students might incorrectly try to apply the product rule to a subtraction of logarithms or vice versa. Always double-check that you’re using the correct rule for the given operation.
  2. Forgetting the Base: It’s crucial to remember the base of the logarithm. If no base is written, it’s usually assumed to be 10 (common logarithm). However, if the base is different (e.g., natural logarithm with base e), the properties and calculations will be different. Always pay attention to the base!
  3. Arithmetic Errors: Simple arithmetic mistakes can derail your entire solution. Whether it’s miscalculating a cube or making an error in division, these mistakes can lead to an incorrect final answer. Take your time and double-check your calculations, especially when dealing with large numbers.
  4. Incorrect Order of Operations: Just like with any mathematical expression, the order of operations (PEMDAS/BODMAS) matters. Make sure you’re applying the rules of logarithms in the correct sequence. For example, you should apply the power rule before the product or quotient rule.
  5. Skipping Steps: It might be tempting to skip steps to save time, but this can often lead to errors. Writing out each step clearly helps you keep track of your work and makes it easier to spot any mistakes. Remember, clarity is key!

By being mindful of these common mistakes, you can improve your accuracy and confidence in solving logarithmic expressions. Now, let’s wrap up with some final thoughts and additional tips for mastering logarithms.

Final Thoughts and Tips

Solving logarithmic expressions might seem challenging at first, but with practice and a solid understanding of the basic properties, you can become a pro! Remember, the key is to break down complex problems into smaller, manageable steps. Here are a few final tips to help you master logarithms:

  • Practice Regularly: The more you practice, the more comfortable you’ll become with applying the properties of logarithms. Try solving a variety of problems to reinforce your understanding.
  • Review the Properties: Keep the logarithmic properties (product, quotient, power rules) handy and refer to them often. Make sure you understand how and when to apply each rule.
  • Use Examples: Work through solved examples to see how the properties are applied in different situations. This can give you a better sense of the problem-solving process.
  • Check Your Work: Always double-check your steps and calculations to avoid errors. If possible, use a calculator to verify your results.
  • Stay Patient: Don’t get discouraged if you struggle at first. Logarithms can be tricky, but with persistence, you’ll get there. Celebrate your progress and keep learning!

By following these tips and practicing consistently, you’ll be well on your way to mastering logarithms. Remember, math is like a puzzle – each piece fits together to create the whole picture. So, keep exploring, keep learning, and have fun with it! And that's a wrap for today's guide on solving 3 Log54 + 3 Log18 - 3 Log12. I hope you found this breakdown helpful and feel more confident tackling similar problems. Keep up the great work, guys!