Evaluate Logarithmic Expression: Log(c^5 / √(ab^3))

Nov 3, 2025 by Admin 52 views

Hey guys! Today, we're diving into the fascinating world of logarithms. We've got a fun problem to tackle that involves evaluating a logarithmic expression. So, let's jump right in and break it down step-by-step. This might seem intimidating at first, but I promise, with a little understanding of log properties, it's totally manageable. We'll take our time, explain each step, and by the end, you'll feel like a log pro! So grab your thinking caps, and let's get started!

Problem Statement

Before we get into the nitty-gritty of solving this logarithmic problem, let's clearly state what we're dealing with. We're given the following logarithmic values:

log a = 6
log b = -10
log c = -8

Our mission, should we choose to accept it (and we do!), is to find the numerical value of the following logarithmic expression:

log(c^5 / √(ab^3))

This expression looks a bit complex, right? Don't worry! We're going to use some fundamental properties of logarithms to simplify it and make it much easier to handle. Think of it like this: we're going to turn a big, scary monster into a cute, cuddly kitten using the magic of log rules. So, are you ready to see how it's done? Let's move on to the next section where we'll unveil the secret weapons in our logarithmic arsenal: the properties of logarithms.

Properties of Logarithms

Okay, before we can even think about directly plugging in those values and crunching numbers, we need to arm ourselves with the right tools. And in this case, our tools are the properties of logarithms. These properties are like the secret sauce that makes simplifying logarithmic expressions a breeze. There are a few key properties we'll be using, so let's break them down:

Power Rule: This rule states that log_b(x^p) = p * log_b(x). In simpler terms, if you have an exponent inside a logarithm, you can bring that exponent down and multiply it by the logarithm. This is going to be super handy for dealing with that c^5 term.
Quotient Rule: This one says that log_b(x / y) = log_b(x) - log_b(y). So, when you're taking the logarithm of a quotient (a division), you can split it into the difference of two logarithms. This will help us separate the numerator (c^5) from the denominator (√(ab^3)).
Product Rule: This rule tells us that log_b(x * y) = log_b(x) + log_b(y). Similar to the quotient rule, but for products! If you have the logarithm of a product, you can break it into the sum of individual logarithms. This will be essential for dealing with the ab^3 inside the square root.
Root Rule: This is actually a special case of the power rule, but it's worth highlighting. Remember that a square root is the same as raising something to the power of 1/2. So, √(x) is the same as x^(1/2). This means we can rewrite the square root in our expression as a fractional exponent and then use the power rule. Pretty neat, huh?

These properties are the keys to unlocking this problem. We're going to use them like puzzle pieces, carefully fitting them together to simplify our expression. In the next section, we'll start applying these properties step-by-step, so you can see exactly how they work in action. Get ready to witness the magic of logarithms!

Applying Logarithmic Properties

Alright, guys, let's get our hands dirty and start applying those logarithmic properties we just discussed! Our goal here is to take that complex expression, log(c^5 / √(ab^3)), and break it down into smaller, more manageable pieces. We'll be like log ninjas, slicing and dicing this expression with our awesome property skills.

First up, let's tackle that division using the Quotient Rule. This rule lets us split the logarithm of a fraction into the difference of two logarithms. Applying it to our expression, we get:

log(c^5 / √(ab^3)) = log(c^5) - log(√(ab^3))

See? We've already made progress! The division is gone, and we now have two separate logarithmic terms. Now, let's focus on each of these terms individually. The first term, log(c^5), looks like a perfect candidate for the Power Rule. Remember, this rule allows us to bring the exponent down as a multiplier. So, we can rewrite log(c^5) as:

log(c^5) = 5 * log(c)

Excellent! We've simplified the first term significantly. Now, let's turn our attention to the second term, log(√(ab^3)). This one looks a bit trickier, but don't worry, we've got this! The first thing we need to do is deal with that square root. Remember from our discussion of properties that a square root is the same as raising something to the power of 1/2. So, we can rewrite log(√(ab^3)) as:

log(√(ab^3)) = log((ab³⁾(1/2))

Now, we can use the Power Rule again to bring down that exponent of 1/2:

log((ab³⁾(1/2)) = (1/2) * log(ab^3)

We're getting there! Now we have log(ab^3) inside the parentheses. This looks like a job for the Product Rule! This rule lets us split the logarithm of a product into the sum of individual logarithms. Applying it, we get:

(1/2) * log(ab^3) = (1/2) * [log(a) + log(b^3)]

Don't forget those brackets! We need to make sure that the 1/2 is multiplied by the entire expression inside. We're almost there, guys! We just have one more small simplification to make. Notice that log(b^3) is another opportunity to use the Power Rule. Let's bring that exponent of 3 down:

(1/2) * [log(a) + log(b^3)] = (1/2) * [log(a) + 3 * log(b)]

Woohoo! We've successfully broken down our original expression into a sum of individual logarithmic terms, each multiplied by a constant. That was quite a journey, wasn't it? In the next section, we'll finally plug in those given values for log(a), log(b), and log(c) and calculate the final answer. You've done a fantastic job so far! Let's finish strong!

Substituting Values and Calculating

Okay, guys, this is the moment we've been working towards! We've successfully simplified our logarithmic expression using all those fancy properties, and now it's time for the grand finale: plugging in the given values and calculating the numerical result. This is where the rubber meets the road, and we see all our hard work pay off.

Let's remind ourselves what we've got. We started with log(c^5 / √(ab^3)) and, after a series of transformations, arrived at the following simplified expression:

5 * log(c) - (1/2) * [log(a) + 3 * log(b)]

And we were given these values:

log a = 6
log b = -10
log c = -8

Now, it's just a matter of substituting these values into our simplified expression. Let's do it!

5 * log(c) - (1/2) * [log(a) + 3 * log(b)] = 5 * (-8) - (1/2) * [6 + 3 * (-10)]

See? We've replaced log(a) with 6, log(b) with -10, and log(c) with -8. Now we just need to carefully perform the arithmetic. Let's start with the multiplication inside the brackets:

5 * (-8) - (1/2) * [6 + 3 * (-10)] = 5 * (-8) - (1/2) * [6 + (-30)]

Next, let's simplify inside the brackets further:

5 * (-8) - (1/2) * [6 + (-30)] = 5 * (-8) - (1/2) * [-24]

Now, let's do the multiplication outside the brackets:

5 * (-8) - (1/2) * [-24] = -40 - (-12)

Remember that subtracting a negative is the same as adding a positive. So, we have:

-40 - (-12) = -40 + 12

Finally, let's add those numbers together:

-40 + 12 = -28

And there you have it! We've successfully evaluated the logarithmic expression. The numerical value of log(c^5 / √(ab^3)) is -28.

Conclusion

Alright, guys, give yourselves a pat on the back! We've conquered a challenging logarithmic problem together. We started with a complex expression, log(c^5 / √(ab^3)), armed ourselves with the properties of logarithms, and systematically broke it down step-by-step. We then substituted the given values for log a, log b, and log c and, after some careful arithmetic, arrived at our final answer: -28.

Remember, the key to tackling logarithmic problems (or any math problem, really) is to break them down into smaller, more manageable steps. Don't be intimidated by the complexity of the initial expression. Focus on applying the rules and properties you know, and you'll be surprised at how quickly things simplify. The power rule, quotient rule, and product rule are your best friends when dealing with logs. Keep practicing with these properties, and you'll become a log-solving master in no time!

I hope this walkthrough has been helpful and has made you feel more confident in your ability to work with logarithms. Keep up the great work, and I'll see you in the next math adventure!

If you have any questions or want to explore more logarithmic problems, feel free to leave a comment below. Let's keep the learning going! You've got this!