Expanding Logarithms: Which Expression Is Correct?

Hey guys! Let's dive into the fascinating world of logarithms and their expanded forms. In this article, we're going to break down how to expand a logarithm and identify the correct expression among a few options. Logarithms can seem a bit daunting at first, but once you grasp the basic rules, expanding them becomes a breeze. We'll walk through the key properties and apply them step-by-step to figure out the right answer. So, buckle up and let's get started!

Understanding Logarithms

Before we jump into the problem, let's quickly recap what logarithms are and the fundamental rules that govern them. At its core, a logarithm is the inverse operation to exponentiation. Think of it this way: if we have an equation like b^y = x, the logarithmic form of this equation is log_b(x) = y. Here, b is the base, x is the argument, and y is the exponent. Understanding this relationship is crucial for manipulating logarithmic expressions.

The main keyword here is expanding logarithms, and we need to understand the rules that allow us to do this. There are three primary rules we'll be using:

1. Product Rule: The logarithm of a product is the sum of the logarithms. Mathematically, this is expressed as log_b(mn) = log_b(m) + log_b(n).
2. Quotient Rule: The logarithm of a quotient is the difference of the logarithms. In mathematical terms, this is log_b(m/n) = log_b(m) - log_b(n).
3. Power Rule: The logarithm of a number raised to a power is the power times the logarithm of the number. This can be written as log_b(m^p) = p * log_b(m).

These rules are our bread and butter when it comes to expanding logarithmic expressions. They allow us to take a complex logarithm and break it down into simpler parts. Remember these rules, guys, because we'll be using them extensively in our problem.

The Problem at Hand

Now, let's get to the specific question we're tackling. We need to choose the expression that correctly shows the expanded form of a given logarithm. We have four options to consider:

A. $10 ext{ln } x+ ext{ln } \sqrt{x-1}+ ext{ln } (3 x-8)$ B. $10 ext{ln } x+\frac{1}{2} ext{ln } (x-1)-\text{ln } (3 x-8)$ C. $\frac{1}{2} \text{ln } x^{10}(x-1)-\text{ln } (3 x-8)$ D. None of the above

To solve this, we need to reverse-engineer the expansion process. We'll start by looking at the options and trying to figure out what the original logarithmic expression might have been. This involves applying the rules we just discussed in reverse. Think of it like detective work – we're using the clues to uncover the original logarithm. The key to correctly expanding logarithms is to carefully apply the product, quotient, and power rules in the correct order.

Breaking Down the Options

Let's analyze each option one by one to see which one correctly represents the expanded form of a logarithm. This is where we put our detective hats on and start piecing together the clues.

Option A: $10 ext{ln } x+\text{ln } \sqrt{x-1}+\text{ln } (3 x-8)$

This option involves the addition of three logarithmic terms. Remembering our product rule, addition of logarithms corresponds to multiplication inside the logarithm. So, we can think of this as:

$\text{ln}(x^{10}) + \text{ln}(\sqrt{x-1}) + \text{ln}(3x-8)$

Using the power rule in reverse on the first term and rewriting the square root as a power, we get:

$\text{ln}(x^{10} * \sqrt{x-1} * (3x-8))$

This gives us a potential original expression. Let's keep this in mind and move on to the next option.

Option B: $10 ext{ln } x+\frac{1}{2} \text{ln } (x-1)-\text{ln } (3 x-8)$

This option is a bit more interesting because it involves both addition and subtraction. The addition part is similar to what we saw in option A, but the subtraction introduces the quotient rule. We can rewrite this as:

$\text{ln}(x^{10}) + \text{ln}((x-1)^{\frac{1}{2}}) - \text{ln}(3x-8)$

Combining the first two terms using the product rule and then applying the quotient rule, we get:

$\text{ln}(x^{10} * (x-1)^{\frac{1}{2}}) - \text{ln}(3x-8)$

$\text{ln}(\frac{x^{10} * \sqrt{x-1}}{3x-8})$

This gives us another potential original expression. Option B looks quite promising, but let's examine the remaining options before making a final decision.

Option C: $\frac{1}{2} \text{ln } x^{10}(x-1)-\text{ln } (3 x-8)$

This option looks a bit tricky. The coefficient of $\frac{1}{2}$ outside the first logarithm suggests that the entire expression inside the logarithm is under a square root. Let's rewrite it:

$\text{ln}((x^{10}(x-1))^{\frac{1}{2}}) - \text{ln}(3x-8)$

Now, applying the quotient rule, we get:

$\text{ln}(\frac{\sqrt{x^{10}(x-1)}}{3x-8})$

This is another possible original expression. We've seen three different possibilities so far, so it's crucial to compare them carefully.

Option D: None of the above

This option is always a possibility, so we need to be absolutely sure that one of the other options is correct before we rule this one out. If none of the expressions we derived from options A, B, and C match the original logarithm we're trying to expand, then this is the correct answer.

Identifying the Correct Expression

Now that we've dissected each option, let's bring it all together and identify the expression that correctly shows an expanded logarithm. We've transformed each option back into a single logarithmic expression. To determine the correct answer, we need to compare these expressions and see if they match the original logarithm we were supposed to expand (which, unfortunately, wasn't explicitly provided in the original prompt, but we can infer the likely original expression based on the options).

From our analysis:

• Option A led us to $\text{ln}(x^{10} * \sqrt{x-1} * (3x-8))$
• Option B led us to $\text{ln}(\frac{x^{10} * \sqrt{x-1}}{3x-8})$
• Option C led us to $\text{ln}(\frac{\sqrt{x^{10}(x-1)}}{3x-8})$

Without knowing the initial logarithm, we have to make an educated guess based on the structure of the options. Option B seems to be the most logically expanded form, as it correctly applies both the product and quotient rules. It separates the terms in a way that reflects the power, product, and division operations within the logarithm.

Conclusion

Alright guys, we've journeyed through the world of expanding logarithms! We've revisited the basic rules, dissected each option, and reverse-engineered the expansion process to identify the correct answer. While the original question didn't explicitly state the logarithm we were expanding, our analysis points to Option B as the most likely correct expanded form:

$10 \text{ln } x+\frac{1}{2} \text{ln } (x-1)-\text{ln } (3 x-8)$

This exercise highlights the importance of understanding and applying the fundamental rules of logarithms. Remember the product, quotient, and power rules, and you'll be well-equipped to tackle any logarithm expansion problem that comes your way. Keep practicing, and you'll become a logarithm pro in no time! And always remember, expanding logarithms is all about breaking down complex expressions into simpler, manageable parts. Great job today!