Unlocking Logarithms: Finding The Equivalent Expression

Hey math enthusiasts! Today, we're diving deep into the world of logarithms. Specifically, we'll figure out which expression is equivalent to $\log _c \frac{x^2-1}{5 x}$. Don't worry, it's not as scary as it looks. We'll break it down step-by-step, making sure you grasp the concepts. So, let's get started and unravel this mathematical mystery! We will explore the properties of logarithms to simplify the given expression. The goal is to rewrite the expression using the properties of logarithms, eventually finding the equivalent expression from the given options. Let's start with a foundational understanding of logarithms, so you can confidently tackle these types of problems.

Understanding the Basics: Logarithms and Their Properties

Alright, before we jump into the problem, let's get our bearings with a quick recap on logarithms and their core properties. Think of logarithms as the opposite of exponents. If you have an equation like $c^y = z$, the logarithmic form would be $\log_c z = y$. The 'c' here is the base, 'z' is the number you're taking the logarithm of, and 'y' is the exponent. Cool, right? Now, the real magic happens when we talk about the properties of logarithms. These are the rules that let us simplify and manipulate logarithmic expressions. These rules are super helpful for solving our main problem.

Firstly, we have the product rule: $\log_c (a * b) = \log_c a + \log_c b$. This tells us that the logarithm of a product is the sum of the logarithms. Next up, we have the quotient rule: $\log_c (a / b) = \log_c a - \log_c b$. This shows that the logarithm of a quotient is the difference of the logarithms. Lastly, the power rule: $\log_c (a^n) = n * \log_c a$. This rule lets us deal with exponents within the logarithm. These are our key weapons in the battle of simplification. Keep these in mind, as we are going to use them to break down the given expression. Mastering these properties is key to solving the main problem. Keep in mind that understanding these properties will make solving the problem much easier. Don't worry if it's not crystal clear at first. We'll clarify these concepts as we work through the problem. Don't worry; we are going to use these properties to solve our main problem. These properties are the tools we need to simplify complex logarithmic expressions and find our answer.

Diving into the Problem

So, our mission is to simplify $\log _c \frac{x^2-1}{5 x}$. The first thing we should notice is the fraction inside the logarithm. This is where the quotient rule comes in handy. Remember, $\log_c (a / b) = \log_c a - \log_c b$. Applying this rule to our problem, we get: $\log _c (x^2-1) - \log _c (5x)$. Now that we have addressed the quotient rule, we can simplify this expression. Now, we are one step closer to solving our problem! Now, we need to look closer at each part of the expression to see if we can simplify it further. We've got $\log _c (x^2-1) - \log _c (5x)$. Notice that $5x$ can be broken down using the product rule. The expression inside the second logarithm, $5x$, is a product of 5 and x. Therefore, we can apply the product rule: $\log_c (a * b) = \log_c a + \log_c b$. The second term can be further simplified as $\log _c 5 + \log _c x$. Therefore, the entire expression becomes $\log _c (x^2-1) - (\log _c 5 + \log _c x)$.

Simplifying Further

Let's keep going. We have $\log _c (x^2-1) - (\log _c 5 + \log _c x)$. Notice that $x^2 - 1$ can be factored as a difference of squares: $(x - 1)(x + 1)$. However, factoring this doesn't lead us to a simpler form that matches the answer choices, so we won't go down that road. Now, let's look at the answer choices to see if our simplified form matches any of them. Remember, we simplified our expression to: $\log _c (x^2-1) - (\log _c 5 + \log _c x)$. If we distribute the negative sign in our simplified expression, it becomes: $\log _c (x^2-1) - \log _c 5 - \log _c x$. However, we still do not have a match for the answers. Let's make sure we didn't make any errors. Let's start with our original expression and use all the properties again. We will use the quotient property first: $\log _c \frac{x^2-1}{5 x} = \log _c (x^2 - 1) - \log _c 5x$. Then, the product property can be used for the second term: $\log _c (5x) = \log _c 5 + \log _c x$. Putting it all together, we have: $\log _c (x^2 - 1) - (\log _c 5 + \log _c x) = \log _c (x^2 - 1) - \log _c 5 - \log _c x$. However, this is still not a match. Let's look at the answer choices.

Analyzing the Answer Choices

Okay, let's carefully go through the answer choices to see which one matches our simplified form or can be further simplified to match our expression: $\log _c \frac{x^2-1}{5 x}$.

• A. $\log _c x^2-\log _c 5 x-1$: This one looks promising at first, but the '-1' is a red flag. It doesn't align with the properties of logarithms. Also, remember, it should be $\log _c (x^2 - 1)$ from the quotient property.
• B. $2 \log _c x-\left(\log _c 5+\log _c x\right)-1$: This answer choice seems off because of the -1 and the missing $\log _c (x^2-1)$. This seems not to be the correct answer because it doesn't align with our previous findings. We can tell this is incorrect.
• C. $\log _c\left(x^2-1\right)-\left(\log _c 5+\log _c x\right)$: Hey, this one matches our simplified form! Remember how we used the quotient rule and product rule? This expression is exactly what we got when we applied those rules. This looks like the right answer.
• D. $2 \log _c x-\log _c 1-\log _c 5+\log _c x$: This doesn't seem quite right. First, $\log _c 1$ is always 0, which doesn't fit the original equation, and it doesn't match the form we found. We can tell this is incorrect. We can immediately eliminate this answer because there is an incorrect term in the original expression. Remember, we used the quotient property, and the answer needs to include $x^2 - 1$.

The Final Answer

So, after careful analysis, the correct answer is C. $\log _c\left(x^2-1\right)-\left(\log _c 5+\log _c x\right)$. It perfectly aligns with the properties of logarithms we used to simplify the original expression. Great job, everyone! You've successfully navigated the world of logarithms. Keep practicing, and you'll become a logarithm pro in no time! Remember, break down the problem step-by-step, use the properties, and you'll always find your way to the correct answer. Congratulations on solving this problem! With a solid understanding of the rules, you can tackle similar problems. Keep up the awesome work.

Conclusion

In conclusion, we've successfully simplified the logarithmic expression $\log _c \frac{x^2-1}{5 x}$ by applying the quotient and product rules. We meticulously analyzed each answer choice, confirming that option C, $\log _c\left(x^2-1\right)-\left(\log _c 5+\log _c x\right)$, is the equivalent expression. This problem highlights the importance of understanding and applying logarithmic properties. Remember to practice these concepts regularly to strengthen your skills. Keep exploring the fascinating world of mathematics, and never stop learning! We hope you enjoyed this journey into the world of logarithms and found this explanation helpful. Keep up the excellent work, and always remember to break down complex problems into manageable steps. Now, go forth and conquer more mathematical challenges!