Log Tables: Evaluating Expressions Made Easy

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Log Tables: Evaluating Expressions Made Easy

Hey guys! Today, we're diving into the fascinating world of logarithmic tables and how they can make evaluating complex expressions a breeze. We'll specifically tackle two expressions: (i) (13.11 * 123) / (36 + 61) and (ii) (543.5 + 28.38) / 293.6. So, buckle up and let's get started!

Understanding Logarithmic Tables

Before we jump into solving the expressions, let's quickly recap what logarithmic tables are and why they're so useful. Logarithmic tables, often called log tables, are tools that help us find the logarithm of a number to a specific base (usually base 10). They simplify complex calculations by transforming multiplication into addition and division into subtraction. This is based on the fundamental properties of logarithms, which state that:

  • log(a * b) = log(a) + log(b)
  • log(a / b) = log(a) - log(b)

The beauty of log tables lies in their ability to break down complex arithmetic operations into simpler ones. This was particularly useful in the pre-calculator era, and even today, they offer a valuable way to understand the mechanics of numerical computation.

Log tables typically consist of two main parts: the characteristic and the mantissa. The characteristic is the integer part of the logarithm and is determined by the position of the decimal point in the number. The mantissa is the decimal part of the logarithm and is found in the log table itself. To effectively use log tables, you need to be comfortable identifying both the characteristic and the mantissa for a given number.

So why bother learning about log tables in the age of calculators? Well, understanding log tables gives you a deeper insight into the nature of logarithms and how they work. It's like knowing the inner workings of an engine instead of just knowing how to drive a car. Plus, it's a fantastic skill to have if you ever find yourself in a situation where you don't have access to a calculator. Trust me; it's happened to the best of us!

Evaluating Expression (i): (13.11 * 123) / (36 + 61)

Let's kick things off by evaluating the first expression: (13.11 * 123) / (36 + 61). We'll break this down step-by-step, using log tables to make the process smoother than butter.

Step 1: Simplify the Denominator

First things first, let's simplify the denominator: 36 + 61 = 97. Now our expression looks like this: (13.11 * 123) / 97.

Step 2: Apply Logarithms

Now comes the fun part! We'll apply logarithms to both the numerator and the denominator. Remember, the goal is to transform multiplication into addition and division into subtraction. So, we have:

log[(13.11 * 123) / 97] = log(13.11 * 123) - log(97)

Using the property log(a * b) = log(a) + log(b), we can further break down the expression:

log(13.11 * 123) - log(97) = log(13.11) + log(123) - log(97)

Step 3: Find the Logarithms Using Log Tables

Time to dust off those log tables! We need to find the logarithms of 13.11, 123, and 97.

  • For log(13.11):
    • The characteristic is 1 (since 13.11 is between 10 and 100).
    • Look up 13 in the log table, and find the value corresponding to 1 under the column for the first decimal place (0.1). Then, look for the mean difference for 1 in the next column. Add these values to get the mantissa. You'll find the mantissa to be approximately 0.1176.
    • So, log(13.11) ≈ 1.1176
  • For log(123):
    • The characteristic is 2 (since 123 is between 100 and 1000).
    • Look up 12 in the log table, and find the value corresponding to 3. You'll find the mantissa to be approximately 0.0899.
    • So, log(123) ≈ 2.0899
  • For log(97):
    • The characteristic is 1 (since 97 is between 10 and 100).
    • Look up 97 in the log table, and find the value corresponding to 0. You'll find the mantissa to be approximately 0.9868.
    • So, log(97) ≈ 1.9868

Step 4: Perform the Arithmetic

Now we have all the logarithmic values we need. Let's plug them into our expression:

log(13.11) + log(123) - log(97) ≈ 1.1176 + 2.0899 - 1.9868

Performing the arithmetic, we get:

  1. 1176 + 2.0899 - 1.9868 ≈ 1.2207

Step 5: Find the Antilogarithm

The final step is to find the antilogarithm of 1.2207. This will give us the value of the original expression. To do this, we use the antilog table.

  • Look up 0.2207 in the antilog table. You'll find the value to be approximately 1.662.
  • Since the characteristic is 1, we multiply the result by 10^1, which is 10.
  • So, the antilog of 1.2207 ≈ 1.662 * 10 = 16.62

Therefore, (13.11 * 123) / (36 + 61) ≈ 16.62

Evaluating Expression (ii): (543.5 + 28.38) / 293.6

Alright, let's move on to the second expression: (543.5 + 28.38) / 293.6. We'll follow a similar approach, using log tables to simplify the calculations.

Step 1: Simplify the Numerator

First, we simplify the numerator: 543.5 + 28.38 = 571.88. Now our expression looks like this: 571.88 / 293.6.

Step 2: Apply Logarithms

Next, we apply logarithms to both the numerator and the denominator. Remember, division transforms into subtraction:

log(571.88 / 293.6) = log(571.88) - log(293.6)

Step 3: Find the Logarithms Using Log Tables

Let's find the logarithms of 571.88 and 293.6 using our trusty log tables.

  • For log(571.88):
    • The characteristic is 2 (since 571.88 is between 100 and 1000).
    • Look up 57 in the log table, find the value corresponding to 1, and then interpolate for 0.88. The mantissa is approximately 0.7573.
    • So, log(571.88) ≈ 2.7573
  • For log(293.6):
    • The characteristic is 2 (since 293.6 is between 100 and 1000).
    • Look up 29 in the log table, find the value corresponding to 3, and then interpolate for 0.6. The mantissa is approximately 0.4678.
    • So, log(293.6) ≈ 2.4678

Step 4: Perform the Arithmetic

Now, let's plug the logarithmic values into our expression:

log(571.88) - log(293.6) ≈ 2.7573 - 2.4678

Performing the subtraction, we get:

  1. 7573 - 2.4678 ≈ 0.2895

Step 5: Find the Antilogarithm

Finally, we find the antilogarithm of 0.2895 to get the value of the original expression.

  • Look up 0.2895 in the antilog table. You'll find the value to be approximately 1.947.
  • Since the characteristic is 0, we multiply the result by 10^0, which is 1.
  • So, the antilog of 0.2895 ≈ 1.947 * 1 = 1.947

Therefore, (543.5 + 28.38) / 293.6 ≈ 1.947

Why Use Log Tables in the Modern Age?

You might be thinking, "Okay, this is cool and all, but why bother with log tables when we have calculators and computers that can do these calculations in milliseconds?" That's a fair question! While it's true that technology has made these tools less necessary for everyday calculations, understanding how log tables work offers some significant advantages:

  1. Deeper Understanding of Logarithms: Using log tables helps you grasp the fundamental principles behind logarithms. You're not just pressing buttons; you're actively engaging with the mathematical concepts.
  2. Historical Context: Log tables played a crucial role in scientific and engineering calculations for centuries. Understanding them gives you a sense of the history of computation.
  3. Backup Skill: What if you're in a situation where you don't have access to a calculator? Knowing how to use log tables can be a lifesaver.
  4. Enhancing Estimation Skills: Working with log tables encourages you to estimate and approximate values, which is a valuable skill in its own right.
  5. Educational Value: Log tables are an excellent tool for teaching mathematical concepts, particularly in the areas of algebra and trigonometry.

Conclusion

So there you have it! We've successfully evaluated two expressions using log tables. It might seem a bit tedious at first, but with practice, you'll get the hang of it. Remember, the key is to break down the problem into smaller steps, apply the logarithmic properties, and use the tables systematically. Mastering log tables not only allows you to perform complex calculations but also gives you a deeper appreciation for the elegance and power of mathematics.

Whether you're a student, a math enthusiast, or just someone who loves learning new things, I hope this guide has been helpful and insightful. Keep exploring, keep learning, and most importantly, keep having fun with math! And hey, if you ever find yourself stranded on a desert island with only a log table, you'll be the hero of the day. 😉