3 Ways to Use Logarithmic Tables

2024 Author: Samantha Chapman | [email protected]. Last modified: 2023-12-17 09:13

Before computers and calculators, logarithms were quickly calculated using logarithmic tables. These tables can still be useful for quickly calculating them or multiplying large numbers once you understand how to use them.

Steps

Method 1 of 3: Read a Logarithmic Table

Step 1. Learn the definition of logarithm

10² = 100. 10³ = 1000. Powers 2 and 3 are the logarithms to base 10, of 100 and 1000. In general, a^b = c can be rewritten as log_toc = b. Thus, saying "ten to two is 100" is equivalent to saying "the logarithm to base 10 of 100 is two". Logarithmic tables are in base 10, so a must always be 10.

• Multiply two numbers by adding their powers. For example: 10² * 10³ = 10⁵, or 100 * 1000 = 100,000.
• The natural logarithm, represented by "ln", is the logarithm to the base "e", where "e" is the constant 2, 718. It is a number widely used in several areas of mathematics and physics. You can use tables relative to the natural logarithm in the same way that you use base 10 ones.

Step 2. Identify the characteristic of the number whose natural logarithm you want to find

15 is between 10 (10¹) and 100 (10²), so its logarithm will be between 1 and 2, and will therefore be "1, something". 150 is between 100 (10²) and 1000 (10³), so its logarithm will be between 2 and 3, and will be "2, something". That "something" is called a mantissa; this is what you find in the logarithmic table. What stands before the decimal point (1 in the first example, 2 in the second) is the characteristic.

Step 3. Swipe your finger to the right row using the leftmost column

This column will show the first two decimal places of the number you are looking for - for some larger boards even three. If you want to find the logarithm of 15, 27 in a base 10 table, go to the line containing 15. If you want to find the log of 2, 577, go to the line containing 25.

• In some cases the numbers in the row will have decimal points, so you will look for 2, 5 rather than 25. You can ignore this decimal point, as it will not affect the result.
• Also ignore any decimal places of the number you are looking for the logarithm for, as the mantissa of the logarithm of 1, 527 is no different than that of 152, 7.

Step 4. In the appropriate row, slide your finger to the correct column

This column will be the one with the first of the decimal digits of the number as heading. For example, if you want to find the logarithm of 15, 27, your finger will be on the row with 15. Scroll your finger to column 2. You will be pointing to the number 1818. Make a note of it.

Step 5. If your table also has the tabular differences, swipe your finger between the columns until you reach the one you want

For 15, 27, the number is 7. Your finger is currently on row 15 and column 2. Scroll to row 15 and tabular difference 7. You will be pointing to number 20. Write it down.

Step 6. Add up the numbers obtained in the previous two steps

For 15, 27, you get 1838. That's the mantissa of the log of 15, 27.

Step 7. Add the feature

Since 15 is between 10 and 100 (10¹ and 10²), the log of 15 must be between 1 and 2, so "1, something", so the characteristic is 1. Combine the characteristic with the mantissa. You will find that the log of 15, 27 is 1, 1838.

Method 2 of 3: Find the Anti-Log

Step 1. Understanding the anti-log table

Use this table when you know the logarithm of a number, but not the number itself. In formula 10 = x, n is the logarithm, to base 10, of x. If you have x, find n using logarithmic tables. If you have n, find x using the anti-log table.

Anti-log is also known as an inverse logarithm

Step 2. Write the feature

It is the number before the decimal point. If you are looking for the anti-log of 2, 8699, the feature is 2. Remove it momentarily from the number you are looking at, but be sure to write it down so you don't forget it - it will be important later on.

Step 3. Find the line that corresponds to the first part of the mantissa

In 2, 8699, the mantissa is ".8699". Most inverse tables, like many logarithmic tables, have two numbers in the leftmost column, so swipe down to ".86".

Step 4. Scroll to the column containing the next mantissa number

For 2, 8699, scroll down to the row with ", 86" and find the intersection with column 9. There should be 7396. Note that.

Step 5. If your table also has tabular differences, swipe the column until you find the next digit of the mantissa

Make sure you stay on the same line. In this case, you will scroll down to the last column, 9. The intersection of row ", 86" and the tabular difference 9 is 15. Make a note of this.

Step 6. Add the two numbers from the previous steps

In our example, they are 7396 and 15. Add them to get 7411.

Step 7. Use the feature to place the decimal point

Our characteristic was 2. This means the answer is between 10² and 10³, or between 100 and 1000. For the number 7411 to be between 100 and 1000, the decimal point must go after the third digit, so that the number is on the order of 700 instead of 70, which is too small, or 7000, which it is too big. So the final answer is 741, 1.

Method 3 of 3: Multiplying Numbers Using Logarithmic Tables

Step 1. Learn to multiply numbers using their logarithms

We know that 10 * 100 = 1000. Written in terms of powers (or logarithms), 10¹ * 10² = 10³. We also know that 1 + 2 = 3. In general, 10^x * 10^y = 10^{x + y}. So the sum of the logarithms of two different numbers is the logarithm of the product of those two numbers. We can multiply two numbers with the same base by adding their powers.

Step 2. Find the logarithms of the two numbers you want to multiply

Use the previous method to calculate them. For example, if you need to multiply 15, 27 and 48, 54, you need to find the log of 15, 27 which is 1.1838 and the log of 48, 54 which is 1.6861.

Step 3. Add the two logarithms to find the logarithm of the solution

In this example, you add 1, 1838 and 1, 6861 to get 2, 8699. This number is the logarithm of your answer.

Step 4. Check the anti-logarithm of the result based on the procedure described in the previous step

You can do this by finding the number in the table as close as possible to the mantissa of this number (8699). However, the most effective method is to use the anti-log table. In this example, you will get 741, 1.

Advice

• Always do the math on paper and not in mind, as these complicated numbers can mislead you.
• Read the page header carefully. A logarithmic table has about 30 pages and using the wrong one will lead you to the wrong answer.

Warnings

• Make sure you are reading from the same line. In some cases, you may get confused due to very thick writing.
• Use the advice given in this article for base 10 logging, and make sure the numbers you are using are in decimal, or scientific notation, format.
• Many tables are accurate only up to the third or fourth digit. If you find the anti-log of 2.8699 using a calculator, the answer will round up to 741.2, but the answer you get using logarithmic tables will be 741.1. This is given to rounding in the tables. If you need a more precise answer, use a calculator or other method.