Solving $\[\log_{4} \left(8 \log_{2} 256\right)\]^{2}$

by ADMIN 57 views

Hey guys! Today, we're diving deep into a fascinating math problem: finding the value of $log⁡4(8log⁡2256){\log_{4} \left(8 \log_{2} 256\right)}^{2}$. This might look intimidating at first, but don't worry, we'll break it down step by step. We'll explore the core concepts of logarithms, simplify the expression systematically, and arrive at the final answer. So, grab your calculators (or just your thinking caps!), and let's get started!

Understanding Logarithms

Before we jump into solving the problem, let's refresh our understanding of logarithms. At its heart, a logarithm answers the question: "To what power must we raise the base to get a certain number?" Mathematically, we express this as:

log⁡ba=x\log_{b} a = x

This reads as "the logarithm of a to the base b is x." It means that b raised to the power of x equals a. Or, in equation form:

bx=ab^{x} = a

For example, $\log_{2} 8 = 3$ because 2 raised to the power of 3 equals 8 ($2^{3} = 8$). The base is the number we're raising to a power (2 in this case), and the argument is the number we want to get (8 in this case). Understanding this fundamental relationship is crucial for simplifying logarithmic expressions.

Key Logarithmic Properties

To effectively solve our problem, we need to be familiar with some key logarithmic properties. These properties allow us to manipulate and simplify complex expressions. Here are a few that we'll use:

  1. Power Rule: $\log_{b} (a^{c}) = c \log_{b} a$ - This rule states that the logarithm of a number raised to a power is equal to the power times the logarithm of the number.
  2. Change of Base Rule: $\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$ - This rule allows us to change the base of a logarithm, which can be useful when dealing with different bases in the same expression.
  3. Logarithm of the Base: $\log_{b} b = 1$ - The logarithm of the base to itself is always 1.
  4. Logarithm of 1: $\log_{b} 1 = 0$ - The logarithm of 1 to any base is always 0.

These properties are our tools for dissecting and simplifying the given expression. We'll use them strategically to break down the problem into manageable steps.

Breaking Down the Expression: $log⁡4(8log⁡2256){\log_{4} \left(8 \log_{2} 256\right)}^{2}$

Now, let's tackle the expression $log⁡4(8log⁡2256){\log_{4} \left(8 \log_{2} 256\right)}^{2}$ step by step. Our goal is to simplify it from the inside out. The innermost part of the expression is $\log_{2} 256$, so that's where we'll start. Remember, we're trying to figure out what power we need to raise 2 to in order to get 256.

Step 1: Simplifying $\log_{2} 256$

We need to express 256 as a power of 2. If you know your powers of 2, you might recognize that $256 = 2^{8}$. If not, you can find this by repeatedly multiplying 2 by itself until you reach 256. Therefore,

log⁡2256=log⁡2(28)\log_{2} 256 = \log_{2} (2^{8})

Now, we can use the power rule of logarithms: $\log_{b} (a^{c}) = c \log_{b} a$. Applying this rule, we get:

log⁡2(28)=8log⁡22\log_{2} (2^{8}) = 8 \log_{2} 2

We also know that $\log_{b} b = 1$, so $\log_{2} 2 = 1$. Therefore:

8log⁡22=8∗1=88 \log_{2} 2 = 8 * 1 = 8

So, we've simplified the innermost part: $\log_{2} 256 = 8$.

Step 2: Substituting and Simplifying Further

Now we can substitute this value back into the original expression:

${\log_{4} \left(8 \log_{2} 256\right)}$^{2} = ${\log_{4} (8 * 8)}$^{2}

Simplifying inside the parentheses:

${\log_{4} (64)}$^{2}

Now we need to find $\log_{4} 64$. We're asking, "To what power must we raise 4 to get 64?" We know that $4^{3} = 64$, so:

log⁡464=3\log_{4} 64 = 3

Step 3: Final Calculation

Substituting this back into our expression, we have:

${\log_{4} (64)}$^{2} = 3^{2}

Finally, we calculate the square:

32=93^{2} = 9

Therefore, the value of $log⁡4(8log⁡2256){\log_{4} \left(8 \log_{2} 256\right)}^{2}$ is 9.

Answer: B) 9

So, the correct answer is B) 9. We successfully navigated this logarithmic expression by breaking it down into smaller, manageable steps. Remember the key properties of logarithms, and you'll be able to tackle similar problems with confidence!

Key Takeaways and Strategies for Solving Logarithmic Problems

Throughout this problem-solving journey, we've uncovered some crucial strategies for handling logarithmic expressions. These takeaways will not only help you with similar questions but also enhance your overall mathematical toolkit. Let's recap the key steps and insights:

  1. Start from the Innermost Expressions: Complex expressions often have nested operations. Just like peeling an onion, start simplifying from the innermost layers. In our case, we began with $\log_{2} 256$, the expression nestled deep within the larger problem.
  2. Master the Core Logarithmic Properties: The power rule, change of base rule, and understanding logarithms of the base and 1 are your best friends. Memorizing and understanding these properties allows you to manipulate and simplify expressions efficiently. Practice applying these rules in various contexts to solidify your understanding.
  3. Express Numbers as Powers of the Base: Identifying and expressing numbers as powers of the base is a game-changer. For instance, recognizing that $256 = 2^{8}$ and $64 = 4^{3}$ significantly simplified our calculations. This skill comes with practice and familiarity with common powers.
  4. Substitute Strategically: As you simplify parts of the expression, substitute the results back into the original equation. This keeps the problem organized and prevents confusion. It's like putting pieces of a puzzle together—each simplified part contributes to the bigger picture.
  5. Don't Be Afraid to Break It Down: Complex problems can feel overwhelming, but remember that they're often just a series of smaller, manageable steps. Break the problem down into smaller chunks, solve each one individually, and then combine the results. This approach makes the problem less daunting and more approachable.
  6. Practice Makes Perfect: Like any mathematical skill, solving logarithmic problems requires practice. Work through various examples, challenge yourself with different complexities, and review your solutions. The more you practice, the more comfortable and confident you'll become.

By applying these strategies, you'll not only ace your exams but also develop a deeper understanding of logarithms and their applications. Math isn't just about memorizing formulas; it's about problem-solving and critical thinking. So, keep exploring, keep practicing, and keep pushing your mathematical boundaries!

Additional Practice Problems

Want to put your newfound skills to the test? Here are a few practice problems similar to the one we just tackled. Try solving them using the strategies we discussed, and don't hesitate to revisit the explanation if you get stuck.

  1. Evaluate: $log⁡3(9log⁡28){\log_{3} \left(9 \log_{2} 8\right)}^{2}$
  2. Simplify: $\log_{5} 125 + \log_{2} 32 - \log_{3} 27$
  3. Find the value of: $log⁡2(18)+log⁡381{\log_{2} \left(\frac{1}{8}\right) + \log_{3} 81}^{3}$

Working through these problems will further solidify your understanding of logarithms and boost your problem-solving confidence. Remember, the key is to break down the expressions, apply the properties strategically, and practice consistently. Good luck, and happy solving!

Conclusion

Well, guys, we made it! We successfully solved the logarithmic expression $log⁡4(8log⁡2256){\log_{4} \left(8 \log_{2} 256\right)}^{2}$ and learned a ton along the way. From understanding the fundamentals of logarithms to mastering key properties and problem-solving strategies, we've equipped ourselves with the tools to tackle similar challenges. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, keep practicing, keep exploring, and never stop questioning. With a solid understanding of the fundamentals and a dash of persistence, you can conquer any mathematical mountain that comes your way. Keep up the awesome work, and I'll catch you in the next mathematical adventure!