Logarithm Table Calculation: Solve 5.34 X 67.4 / 2.7

Hey guys, ever found yourself staring at a complex calculation and wishing for a magic wand? Well, back in the day, before fancy calculators and smartphones, mathematicians had their own secret weapon: logarithm tables. These handy tools were a lifesaver for simplifying tricky multiplications and divisions. Today, we're going to dive into a classic problem using these tables: evaluating $\frac{5.34 \times 67.4}{2.7}$. We'll break down how to use log tables step-by-step, making this seemingly daunting task super manageable. So, grab your virtual log tables (or just follow along!), and let's get this calculation sorted. We'll be looking at the options A. 1.332, B. 13.32, C. 133.2, D. 1332, and E. 13.320, to see which one is the correct answer after our log table adventure.

Understanding the Power of Logarithms

Alright, let's chat about why logarithm tables were such a big deal, especially for calculations like $\frac{5.34 \times 67.4}{2.7}$. The core idea behind logarithms is that they can turn multiplication into addition and division into subtraction. Think about it: if you have $log(a \times b)$, it's the same as $log(a) + log(b)$. And if you have $log(\frac{a}{b})$, that's equal to $log(a) - log(b)$. This is a game-changer, guys! Instead of wrestling with multiplying large numbers or dividing by decimals, you can just look up the logarithms of those numbers in a table, do some simple addition or subtraction, and then use the table again to find the final answer. It's like having a cheat code for math! For our problem, $\frac{5.34 \times 67.4}{2.7}$, we can rewrite it using logarithms like this:

$log(\frac{5.34 \times 67.4}{2.7}) = log(5.34) + log(67.4) - log(2.7)$

See how that works? We've transformed a multiplication and a division into a sum and a difference. This is the fundamental principle that makes logarithm tables so powerful for simplifying complex arithmetic. It allowed scientists, engineers, and anyone doing heavy calculations to work much faster and with greater accuracy before electronic calculators became commonplace. The process involves finding the characteristic and mantissa for each number, performing the arithmetic on the mantissas, and then using the antilogarithm table to find the result. It requires a bit of practice, but once you get the hang of it, it feels incredibly efficient.

Step-by-Step Logarithm Table Evaluation

Now, let's get down to business and evaluate $\frac{5.34 \times 67.4}{2.7}$ using our logarithm tables. Remember, the goal is to find $log(5.34) + log(67.4) - log(2.7)$. First things first, we need to find the logarithm for each of these numbers. Logarithm tables typically give you the mantissa (the decimal part) and you determine the characteristic (the integer part) based on the number's magnitude.

1. Finding the Logarithm of 5.34:

To find $log(5.34)$, we look at the number 5.34. The characteristic is determined by the position of the decimal point. For numbers greater than or equal to 1, the characteristic is one less than the number of digits before the decimal. So, for 5.34, there's one digit before the decimal, meaning the characteristic is $1 - 1 = 0$. Now, we look up the mantissa for '53' in the '4' column of our logarithm table. Let's assume, for example, that the table gives us a mantissa of approximately $0.7275$. So, $log(5.34) \approx 0.7275$.

2. Finding the Logarithm of 67.4:

For 67.4, there are two digits before the decimal point (6 and 7). Therefore, the characteristic is $2 - 1 = 1$. We then look up the mantissa for '67' in the '4' column. Let's say our table gives us a mantissa of approximately $0.8287$. Thus, $log(67.4) \approx 1.8287$.

3. Finding the Logarithm of 2.7:

For 2.7, there is one digit before the decimal point. The characteristic is $1 - 1 = 0$. We look up the mantissa for '27' in the '0' column (or usually, it's directly under the '27' row if there's no specific column for '0'). Let's assume the mantissa is approximately $0.4314$. So, $log(2.7) \approx 0.4314$.

4. Performing the Addition and Subtraction:

Now we substitute these values back into our logarithmic equation:

$log(\text{Result}) = log(5.34) + log(67.4) - log(2.7)$

$log(\text{Result}) \approx 0.7275 + 1.8287 - 0.4314$

First, add the positive logarithms:

$0.7275 + 1.8287 = 2.5562$

Now, subtract the logarithm of the denominator:

$2.5562 - 0.4314 = 2.1248$

So, we have $log(\text{Result}) \approx 2.1248$. This is the logarithm of our final answer.

Finding the Antilogarithm

We've done the hard part with the logarithms, guys! Now we need to find the antilogarithm of 2.1248 to get our actual result. The antilogarithm is the inverse operation of the logarithm. It tells us what number, when raised to the base of the logarithm (usually 10 for common logarithms), gives us our current value. Our antilog is 2.1248. The characteristic '2' tells us about the magnitude of our final answer, and the mantissa '0.1248' is what we use to look up the number in the antilogarithm table.

1. Using the Antilogarithm Table:

We look for the mantissa '0.1248' in the antilogarithm table. Typically, you'll look for '12' in the first column and '4' in the top row. Let's say the table gives us a value of approximately $1.333$ for the first three digits (0.124). Then, there might be a 'mean difference' column for the last digit '8'. Let's assume the mean difference for '8' is $0.001$. We add these together: $1.333 + 0.001 = 1.334$. So, the number corresponding to the mantissa 0.1248 is approximately 1.334.

2. Applying the Characteristic:

Now, we use the characteristic, which is '2'. The characteristic tells us the position of the decimal point in our final answer. For a positive characteristic $n$, the antilogarithm has $n+1$ digits before the decimal point. Since our characteristic is 2, our answer will have $2 + 1 = 3$ digits before the decimal point.

So, we take our antilog value (1.334) and place the decimal point to get 3 digits before it. This gives us 133.4.

Comparing this to our options:

A. 1.332 B. 13.32 C. 133.2 D. 1332 E. 13.320

Our calculated value of 133.4 is very close to 133.2 (option C). The slight difference is likely due to rounding or minor variations in the specific logarithm tables used. In a multiple-choice question scenario, this would be our clear winner!

Why Logarithm Tables Still Matter (Kind Of!)

So, why are we even bothering with logarithm tables in the age of instant calculations? Well, understanding how they work gives you a deeper appreciation for the mathematics behind computation. It's like understanding how an engine works instead of just driving a car. Plus, these tables were essential tools for generations of scientists and engineers, enabling breakthroughs that shaped our modern world. They taught us the fundamental properties of logarithms and how to manipulate them efficiently. While you might not be carrying a log table around, the principles you learn from using them are still super relevant in higher mathematics and computer science, especially in areas like algorithm analysis and understanding exponential growth.

Moreover, solving problems like $\frac{5.34 \times 67.4}{2.7}$ using log tables forces you to think critically about number properties and precision. It’s a fantastic exercise in mental math and numerical reasoning. It highlights how a clever mathematical tool can simplify complex operations, making them accessible and manageable. So, even though calculators have taken over the heavy lifting, the logic and skills honed by using logarithm tables remain valuable. It's a piece of mathematical history that offers practical insights into numerical methods. It’s a testament to human ingenuity in tackling complex problems with basic tools. The process itself is a mathematical journey, turning a cumbersome calculation into a series of simpler, manageable steps, ultimately leading to a precise result. It's pretty cool when you think about it!