Logarithm Properties: Rewriting Log Base 7 Of 98

Hey guys! Ever get stuck trying to simplify logarithms? Today, we're diving deep into how to rewrite a logarithm, specifically $\log_7 98$, using the awesome product property of logarithms. We'll also figure out what $\log_7 49$ is, all while keeping it fun and easy to understand. So, grab your calculators (or just your brains!) and let's get cracking!

Understanding the Product Property of Logarithms

First things first, let's talk about the product property of logarithms. This is one of those super handy rules that makes dealing with logs way less of a headache. Basically, it says that the logarithm of a product is equal to the sum of the logarithms of the factors. In mathematical terms, if you have $\log_b (xy)$, you can rewrite it as $\log_b x + \log_b y$. See? Much simpler, right? This property is your best friend when you need to break down a complex logarithm into smaller, more manageable pieces. It's like taking a big, scary monster and turning it into a bunch of little, friendly critters. When we apply this to our problem, $\log_7 98$, we need to think about how to express 98 as a product of numbers that might be easier to work with, especially in relation to our base, which is 7. We're given a hint that $\log_7 2 \approx 0.356$, which suggests that the number 2 will be useful. So, how can we get 98 by multiplying numbers, ideally involving 2 and maybe something related to 7? Let's break 98 down. We know 98 is an even number, so it's divisible by 2. $98 \div 2 = 49$. Aha! So, 98 can be written as $2 \times 49$. Now, we can use our product property. Instead of $\log_7 98$, we can write $\log_7 (2 \times 49)$. Applying the product property, this becomes $\log_7 2 + \log_7 49$. This is a huge step because we've transformed a single, slightly awkward logarithm into the sum of two logarithms. One of them, $\log_7 2$, is something we're given information about (its approximate value). The other, $\log_7 49$, looks promisingly simple because 49 is a power of 7. This is exactly what the product property is designed for: to simplify expressions by breaking them down. Remember, the goal is often to isolate terms that are easy to evaluate or terms for which you have given values. In this case, breaking 98 into $2 \times 49$ was the key move, enabled by the product property.

Evaluating $\log_7 49$

Now, let's tackle the second part of our puzzle: what is $\log_7 49$? This one is actually pretty straightforward once you understand the definition of a logarithm. Remember, $\log_b a = c$ means that $b^c = a$. In our case, $\log_7 49 = x$ means we are looking for the exponent $x$ such that $7^x = 49$. Think about it: what power do you need to raise 7 to in order to get 49? We all know (or can quickly figure out) that $7 \times 7 = 49$. And $7 \times 7$ is the same as $7^2$. So, if $7^x = 49$ and $7^2 = 49$, then $x$ must be 2! Therefore, $\log_7 49 = 2$. It's that simple, guys! This is a prime example of how understanding the relationship between exponents and logarithms can make things incredibly easy. The number 49 is a perfect square of our base, 7, which makes its logarithm base 7 a nice, whole number. This is often the case when the argument of the logarithm is a power of the base. If you had $\log_3 81$, you'd ask yourself, "3 to what power equals 81?" Since $3^4 = 81$, then $\log_3 81 = 4$. The same logic applies here. Because $7^2 = 49$, the logarithm $\log_7 49$ tells us that the exponent needed is 2. This evaluation is crucial because it allows us to simplify the expression we derived using the product property.

Rewriting $\log_7 98$ Using the Product Property

Alright, let's put it all together! We started with $\log_7 98$. We identified that 98 can be written as the product of 2 and 49 ($98 = 2 \times 49$). Then, we used the product property of logarithms, which states $\log_b (xy) = \log_b x + \log_b y$. Applying this, we got: $\log_7 98 = \log_7 (2 \times 49) = \log_7 2 + \log_7 49$. This is our rewritten form using the product property. Now, we also figured out that $\log_7 49 = 2$. So, we can substitute that value in: $\log_7 98 = \log_7 2 + 2$. This is the fully simplified expression using the product property and evaluating the part we could. We were given that $\log_7 2 \approx 0.356$. So, we can even approximate the original logarithm: $\log_7 98 \approx 0.356 + 2 = 2.356$. This shows the power of these properties – transforming a seemingly complex number into something much more understandable and calculable. The process involved recognizing the structure of the number inside the logarithm (98), finding factors that relate to the base (7) or are given (like $\log_7 2$), and then applying the appropriate logarithm property. The product property was key here, allowing the transition from a single log term to a sum of log terms. The evaluation of $\log_7 49$ further simplified the expression, making it ready for numerical substitution. It’s a beautiful chain reaction of mathematical logic!

Examining the Options

Now that we've done the heavy lifting, let's look at the multiple-choice options provided to see which one matches our work. We found that $\log_7 98$ can be rewritten as $\log_7 2 + \log_7 49$. Let's check the options:

• A. $7 \log 2+7 \log 49$: This looks like it's trying to use some other properties, maybe the power rule incorrectly, and also it's missing the base 7. The product property gives a sum of logs, not a sum multiplied by a number. So, this one is a definite no.
• B. $\log 7+\log 2+\log 49$: This option is missing the base 7 on all the logarithms, which is a critical part of the problem. Also, if it were $\log_7 7 + \log_7 2 + \log_7 49$, that would be $1 + \log_7 2 + 2$, which simplifies to $3 + \log_7 2$. That's not quite what we got. The product property applies to the argument of the logarithm, not the base itself being multiplied. So, nope.
• C. $\log_7 2+\log_7 49$: This perfectly matches our rewritten form using the product property before we evaluated $\log_7 49$. We rewrote $\log_7 98$ as $\log_7 (2 \times 49)$ and then applied the product rule to get $\log_7 2 + \log_7 49$. Yes, this is it!

So, the correct way to rewrite $\log_7 98$ using the product property is $\log_7 2 + \log_7 49$. And as we discussed, $\log_7 49$ is equal to 2.

Conclusion

Guys, we've successfully navigated the world of logarithm properties! We learned that the product property lets us break down $\log_b (xy)$ into $\log_b x + \log_b y$. We used this to rewrite $\log_7 98$ as $\log_7 2 + \log_7 49$. We also discovered that $\log_7 49$ is simply 2, because $7^2 = 49$. This makes the expression $\log_7 2 + 2$. If we use the given approximation, $\log_7 2 \approx 0.356$, then $\log_7 98 \approx 0.356 + 2 = 2.356$. Remember these properties, especially the product property, as they are fundamental tools in your math arsenal. Keep practicing, and you'll be simplifying logs like a pro in no time! What else can we break down with these awesome rules? Let us know in the comments!