Solving 3 Log₄x = Log₄32 + Log₄2: A Step-by-Step Guide

Hey there, math enthusiasts! Ever stumbled upon a logarithmic equation that seems like a puzzle? Well, you're not alone! Logarithmic equations can appear daunting at first, but with the right approach, they can be cracked open like a coconut. Today, we're going to dive deep into solving the equation 3 log₄x = log₄32 + log₄2. We'll break it down step-by-step, ensuring you not only understand the solution but also the underlying principles. So, grab your thinking caps, and let's get started!

Understanding the Fundamentals of Logarithms

Before we jump into the solution, let's quickly refresh our understanding of logarithms. Logarithms, at their core, are the inverse operation of exponentiation. Think of it this way: if 2³ = 8, then log₂8 = 3. The logarithm (log) tells you what exponent you need to raise the base (in this case, 2) to, in order to get a specific number (8). The '3' in our example is the exponent. This foundational concept is super critical to grasp, guys, because it's the bedrock upon which we build our understanding of more complex logarithmic manipulations. Without a solid understanding of the basic relationship between exponents and logarithms, navigating through equation-solving becomes a much trickier task. Really knowing this relationship is like having the secret decoder ring for logarithmic puzzles.

Logarithms also come with a set of handy properties that make solving equations much easier. Two key properties we'll use today are:

1. The Power Rule: logₐ(xⁿ) = n logₐ(x). This rule allows us to move exponents inside a logarithm to the front as a coefficient, or vice versa.
2. The Product Rule: logₐ(x) + logₐ(y) = logₐ(xy). This rule tells us that the sum of logarithms with the same base is equal to the logarithm of the product of their arguments. Understanding these properties is like having the right tools in your math toolbox – they empower you to simplify and solve equations efficiently.

These properties are not just abstract rules; they are the keys that unlock the solution to logarithmic equations. They allow us to manipulate the equation, combine terms, and ultimately isolate the variable we're trying to solve for. So, as we move through the solution, keep these properties in mind, and you'll see how they come into play.

Step-by-Step Solution: Cracking the Code

Now, let's tackle the equation 3 log₄x = log₄32 + log₄2. We'll proceed step-by-step, showing each manipulation clearly so you can follow along.

Step 1: Applying the Product Rule

Our first move is to simplify the right side of the equation using the product rule of logarithms. We have log₄32 + log₄2. According to the product rule, this can be rewritten as log₄(32 * 2). So, let's do that:

3 log₄x = log₄(32 * 2)

Simplifying the multiplication, we get:

3 log₄x = log₄64

This step is all about condensation. We're taking two separate logarithmic terms and merging them into one, which makes the equation cleaner and easier to work with. Think of it like combining two puzzle pieces into a larger section – you're making progress toward the bigger picture.

Step 2: Applying the Power Rule

Next, we'll use the power rule to move the coefficient '3' in front of the logarithm on the left side. The power rule states that n logₐ(x) = logₐ(xⁿ). Applying this rule, we get:

log₄(x³) = log₄64

This step is like reshaping the equation. By moving the coefficient into the exponent, we're setting the stage for the next crucial step – eliminating the logarithms altogether. It's a clever maneuver that simplifies the equation and brings us closer to the solution.

Step 3: Eliminating the Logarithms

Now comes the exciting part! Since we have logarithms with the same base (base 4) on both sides of the equation, we can simply eliminate them. This is because if logₐ(x) = logₐ(y), then x = y. So, we can rewrite our equation as:

x³ = 64

This is a major breakthrough! We've transformed the logarithmic equation into a simple algebraic equation. It's like reaching the summit of a hill – you can see the path ahead much more clearly now. By dropping the logarithms, we've stripped the equation down to its essential form, making it much easier to solve.

Step 4: Solving for x

To solve for x, we need to find the cube root of 64. What number, when multiplied by itself three times, equals 64? The answer is 4:

x = ∛64 x = 4

And there you have it! We've found the solution. The value of x that satisfies the original equation is 4. This final step is the culmination of all our efforts. It's the moment when the puzzle pieces click into place, and the solution is revealed.

Verification: Ensuring the Solution is Correct

It's always a good practice to verify our solution. Let's plug x = 4 back into the original equation and see if it holds true:

3 log₄(4) = log₄32 + log₄2

We know that log₄(4) = 1 (since 4¹ = 4). Also, we already simplified log₄32 + log₄2 to log₄64, and log₄64 = 3 (since 4³ = 64). So, our equation becomes:

3 * 1 = 3

3 = 3

The equation holds true! This confirms that our solution, x = 4, is indeed correct. Verification is the ultimate safety net. It ensures that our solution is not just a result of correct manipulations but also a true reflection of the equation's inherent relationships.

Conclusion: Mastering Logarithmic Equations

So, guys, we've successfully navigated the logarithmic landscape and solved the equation 3 log₄x = log₄32 + log₄2. We've seen how the properties of logarithms, like the power rule and product rule, are essential tools in simplifying and solving these equations. Remember, the key to mastering logarithmic equations is understanding the fundamentals, applying the properties correctly, and practicing consistently. Keep those thinking caps on, and you'll be conquering logarithmic equations in no time!

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Title: Solving 3 log₄x = log₄32 + log₄2 A Step-by-Step Guide