Evaluating The Expression (ln(π) - 5ln(4))/(ln(π) - 3ln(4)) With A Calculator

Hey guys! Today, we're diving into a mathematical expression that looks a bit intimidating at first glance, but don't worry, we'll break it down step-by-step using a calculator. We're going to evaluate the expression:

$$\frac{\ln \pi - 5 \ln 4}{\ln \pi - 3 \ln 4}$$

This expression involves natural logarithms (ln) and a fraction, so it's important to tackle it methodically. We'll walk through each part of the expression, calculate the values using a calculator, and then combine the results to get our final answer. So, grab your calculators, and let's get started!

Understanding the Expression

Before we jump into the calculations, let's make sure we understand what each part of the expression means. The expression consists of a fraction with a numerator and a denominator. Both the numerator and the denominator involve natural logarithms (ln). A natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is an irrational number approximately equal to 2.71828. Natural logarithms are widely used in mathematics, physics, engineering, and other fields. Understanding these fundamental concepts is crucial for accurately evaluating the expression. So let's take a closer look at each component of the expression. The numerator is composed of two terms: ln(π) and 5 ln(4). The first term, ln(π), requires finding the natural logarithm of the mathematical constant π (pi), which is approximately 3.14159. The second term, 5 ln(4), involves first finding the natural logarithm of 4 and then multiplying the result by 5. This operation follows the logarithmic property that allows constants to be multiplied by logarithms. Similarly, the denominator also comprises two terms: ln(π) and 3 ln(4). The term ln(π) is the same as in the numerator, representing the natural logarithm of π. The second term, 3 ln(4), involves finding the natural logarithm of 4 and then multiplying the result by 3. Again, this is an application of the logarithmic property. By dissecting the expression into its components, we can clearly see the order of operations required to evaluate it. The first step is to find the natural logarithms of π and 4. Next, these logarithms are multiplied by the constants 5 and 3 in the numerator and denominator, respectively. Finally, the results are combined through subtraction, and the entire numerator is divided by the denominator. This systematic approach ensures that we follow the correct mathematical procedure, leading to an accurate final answer. Remember, understanding each part of the expression and the order of operations is vital for successful evaluation. Let's move on to the next section, where we will actually start calculating these values using a calculator.

Calculating ${\ln \pi}$ and ${\ln 4}$

Okay, guys, let's get our hands dirty with some actual calculations! The first step in evaluating our expression is to find the values of ${\ln \pi}$ and ${\ln 4}$. For this, you'll need a calculator that has a natural logarithm function (usually labeled as "ln"). Now, let's start with ${\ln \pi}$. On your calculator, input the value of ${\pi}$ (usually found as a dedicated button or as a second function of another key) and then press the "ln" button. You should get a result close to 1.1447. Remember, we're dealing with irrational numbers, so the decimal representation goes on forever, but we'll round it to four decimal places for our calculations. So,

$$\ln \pi \approx 1.1447$$

Next up, we need to calculate ${\ln 4}$. This one is a bit more straightforward since we're dealing with a whole number. Simply input 4 into your calculator and press the "ln" button. You should get a result close to 1.3863. Again, we'll round to four decimal places for consistency. So,

$$\ln 4 \approx 1.3863$$

Now that we have these two values, we're one step closer to solving the entire expression. It's crucial to write these values down or keep them stored in your calculator's memory, as we'll need them for the next steps. These two logarithms are the building blocks for both the numerator and the denominator of our fraction. Understanding and accurately calculating these values is key to ensuring the final result is correct. A common mistake is to rush through these initial calculations, which can lead to errors later on. Remember, mathematics is a step-by-step process, and each step relies on the accuracy of the previous one. Now, let's take a moment to recap what we've done. We've successfully used a calculator to find the natural logarithms of both ${\pi}$ and 4, and we've recorded these values for further use. With these values in hand, we are well-prepared to move on to the next phase of our calculation, which involves incorporating these logarithms into the rest of the expression. So, grab those values and let's move on to the next section, where we'll use them to evaluate the numerator and denominator separately. We're making good progress, guys! Let's keep it up!

Evaluating the Numerator

Alright, let's tackle the numerator of our expression: ${\ln \pi - 5 \ln 4}$. We've already found that ${\ln \pi \approx 1.1447}$ and ${\ln 4 \approx 1.3863}$. Now, we just need to plug these values into the numerator and do the arithmetic. First, we need to calculate ${5 \ln 4}$. This means we multiply the value we found for ${\ln 4}$ by 5:

$$5 \ln 4 \approx 5 \times 1.3863 = 6.9315$$

Now that we have this value, we can substitute it back into the numerator:

$$\ln \pi - 5 \ln 4 \approx 1.1447 - 6.9315$$

Subtracting these values gives us:

$$1. 1447 - 6.9315 = -5.7868$$

So, the value of the numerator is approximately -5.7868. It's super important to keep track of the negative sign here! A common mistake is to forget the negative sign, which can completely change the final result. This is why paying close attention to detail and double-checking your calculations is crucial in mathematics. Now that we've successfully evaluated the numerator, we're halfway to solving the entire expression. The process we followed here is a great example of how to break down a complex problem into smaller, more manageable parts. By calculating each component separately and then combining the results, we avoid getting overwhelmed and reduce the risk of errors. Remember, guys, math is like building a house – each step is a foundation for the next. Now, before we get too comfortable, let's shift our focus to the denominator. We'll use a similar approach to evaluate it, plugging in the values we calculated earlier and performing the necessary arithmetic. With the numerator under our belt, we're well-equipped to tackle the denominator. So, let's head on over to the next section, where we'll do just that. Keep those calculators handy, and let's get to work!

Evaluating the Denominator

Okay, now let's move on to the denominator of our expression: ${\ln \pi - 3 \ln 4}$. We already know that ${\ln \pi \approx 1.1447}$ and ${\ln 4 \approx 1.3863}$, so we'll use these values just like we did for the numerator. The first step here is to calculate ${3 \ln 4}$. This means multiplying the value of ${\ln 4}$ by 3:

$$3 \ln 4 \approx 3 \times 1.3863 = 4.1589$$

Now we can substitute this back into the denominator:

$$\ln \pi - 3 \ln 4 \approx 1.1447 - 4.1589$$

Subtracting these values, we get:

$$1. 1447 - 4.1589 = -3.0142$$

So, the value of the denominator is approximately -3.0142. Just like with the numerator, it's essential to pay attention to the negative sign. These little details can make a big difference in the final answer. By evaluating the denominator separately, we've kept our calculations organized and reduced the chances of making a mistake. This step-by-step approach is a powerful tool in mathematics, allowing us to tackle complex problems with confidence. We've now successfully calculated both the numerator and the denominator of our expression. That's a huge accomplishment! We're almost at the finish line. Before we move on, let's take a moment to appreciate the work we've done so far. We've broken down a seemingly complicated expression into manageable parts, calculated the necessary logarithms, and evaluated both the numerator and the denominator. This is the essence of problem-solving in mathematics: to simplify, to calculate accurately, and to stay organized. Now, with the values of the numerator and the denominator in hand, we're ready for the final step: dividing the numerator by the denominator. This will give us the final result of our expression. So, let's move on to the next section, where we'll perform this division and wrap up our calculation. Get ready to see the final answer, guys! We're almost there!

Dividing Numerator by Denominator

Alright, guys, we've reached the final step! We've calculated the numerator to be approximately -5.7868 and the denominator to be approximately -3.0142. Now, all that's left to do is divide the numerator by the denominator to get our final answer.

So, we have:

$$\frac{\ln \pi - 5 \ln 4}{\ln \pi - 3 \ln 4} \approx \frac{-5.7868}{-3.0142}$$

Now, let's perform the division using a calculator. Input -5.7868 divided by -3.0142. Remember that dividing a negative number by a negative number results in a positive number, so our final answer should be positive. The result we get is approximately:

$$\frac{-5.7868}{-3.0142} \approx 1.9200$$

Therefore, the value of the expression is approximately 1.9200. We've done it! We've successfully evaluated the entire expression using a calculator. This final step highlights the importance of paying attention to the signs of the numbers. A simple mistake with a negative sign could have thrown off our entire calculation. But we stayed focused, followed the steps carefully, and arrived at the correct answer. Now, let's take a moment to reflect on the entire process. We started with a complex-looking expression involving natural logarithms and fractions. We broke it down into smaller, more manageable parts. We calculated the natural logarithms of ${\pi}$ and 4. We used these values to evaluate the numerator and the denominator separately. And finally, we divided the numerator by the denominator to get our final answer. This is a great example of how mathematical problems can be solved by breaking them down into smaller steps. It's like climbing a ladder – each step gets you closer to the top. And the satisfaction of reaching the top, in this case, finding the final answer, is well worth the effort. So, congratulations, guys! We've successfully navigated this mathematical challenge. Let's move on to the conclusion, where we'll recap our steps and discuss the key takeaways from this exercise.

Conclusion

Woo-hoo! We did it! We successfully evaluated the expression

$$\frac{\ln \pi - 5 \ln 4}{\ln \pi - 3 \ln 4}$$

using a calculator. Our final answer is approximately 1.9200. Let's quickly recap the steps we took to get there:

1. Understood the Expression: We identified the natural logarithms and the fraction, recognizing the order of operations.
2. Calculated ${\ln \pi}$ and ${\ln 4}$: We used a calculator to find that ${\ln \pi \approx 1.1447}$ and ${\ln 4 \approx 1.3863}$.
3. Evaluated the Numerator: We calculated ${5 \ln 4}$, then subtracted it from ${\ln \pi}$ to get approximately -5.7868.
4. Evaluated the Denominator: We calculated ${3 \ln 4}$, then subtracted it from ${\ln \pi}$ to get approximately -3.0142.
5. Divided Numerator by Denominator: We divided -5.7868 by -3.0142 to get our final answer of approximately 1.9200.

This exercise highlights several important concepts in mathematics. First, it demonstrates the importance of understanding the order of operations. We had to perform the multiplications before the subtractions, and we had to evaluate the numerator and denominator separately before dividing. Second, it shows the power of breaking down complex problems into smaller, more manageable steps. By tackling each part of the expression individually, we were able to avoid getting overwhelmed and reduce the risk of errors. Third, it emphasizes the importance of accuracy in calculations. Even a small mistake in one step could have thrown off our entire answer. Finally, it reinforces the use of calculators as a tool for solving mathematical problems. While it's important to understand the underlying concepts, calculators can help us perform complex calculations quickly and accurately. So, what are the key takeaways from this exercise, guys? Well, first and foremost, remember that complex expressions can be tamed by breaking them down into simpler parts. Second, paying close attention to detail is crucial – especially when dealing with negative signs. Third, don't be afraid to use your calculator! It's a powerful tool that can help you solve all sorts of mathematical problems. And last but not least, practice makes perfect! The more you work with these types of expressions, the more comfortable and confident you'll become. So, keep practicing, keep exploring, and keep challenging yourselves. Math is an adventure, and there's always something new to discover. Thanks for joining me on this mathematical journey, guys! I hope you found this explanation helpful and informative. Now go out there and conquer those mathematical challenges!