In mathematics, the** logarithm table** is used to find the value of the logarithmic function. The simplest way to find the value of the given logarithmic function is by using the **log table**.

Given belowÂ is the common log table (i.e., for base 10). This table can give the value of log x (which is also written as log_{10} x) for any x.

The log table mainly has 3 types of columns:

- The main column with numbers from 10 – 99 (all are 2 digit numbers)
- The differences column which shows the differences for the digits 0 – 9
- The mean differences column showing the mean differences from 1 – 9

There are also logarithm tables for base e (which is called the natural logarithm table) and for base 2 (which is called the binary logarithm table).

### How to Use the Log Table?

The logarithm of any number consists of two parts: an integer(whole number) and a fraction after the decimal point.

The integral part is called the **characteristic** and the fractional or decimal part is called the **mantissa**.

**Step 1:** Identify the characteristic part and mantissa part of the given number. For example, if you want to find the value of log_{10} (13.93), first separate the characteristic part and the mantissa part.

Characteristic Part = 13

Mantissa part = 93

**Step 2:** Use a common log table. Now, use row number 13 and check column number 9 and write the corresponding value. So the value obtained is **1430**.

**Step 3:** Use the logarithm table with a mean difference. Slide your finger to the mean difference column number 3 and row number 13, and write down the corresponding value as **10**.

**Step 4:** Add both the values obtained in step 3 and step 4. That is 1430 + 10= 1440. Therefore, the value 1440 is the mantissa part.

**Step 5:** Find the characteristic part. The characteristic of the logarithm of a number is the exponent of 10 in its standard form. You have to express the number in standard form first then identify the exponent of 10.

For example;

Number | Standard Form | Characteristic of Log of Number |
---|---|---|

60.78 | 6.078 Ã— 10^{1} | 1 |

1.562 | 1.562 Ã— 10^{0} | 0 |

783 | 7.83 Ã— 10^{2} | 2 |

In our example, 13.93 expressed in standard form is 1.393 Ã— 10^{1}, therefore, the characteristic is 1

**Step 6:** Finally combine both the characteristic part and the mantissa part, it becomes 1.1440.

Thus, 1.1440 is the index or logarithm of 13.93 to base 10

Then, 13.93 = \( \scriptsize 1.393 \: \times \: 10^1 \\\scriptsize = 10^{0.1440} \: \times \: 10^1 \\ \scriptsize = 10^{0.1440 \: +\: 1} \\ \scriptsize = 10^{1.1440} \)

Let’s say you were asked to find the value of log_{10}139.3 instead. All the steps will be the same as above excpet step 5 and 6. In this case, the standard form is 1.393 Ã— 10^{2}, therefore, the characteristic is 2.

Combining the characteristic and mantissa parts will give 2.1440.

139.3 = \( \scriptsize 1.393 \: \times \: 10^2 \\\scriptsize = 10^{0.1440} \: \times \: 10^2 \\ \scriptsize = 10^{0.1440 \: +\: 2} \\ \scriptsize = 10^{2.1440} \)

### Example 1:Â

Find the common log of each of these numbers using the log table. Also, verify the answers using a calculator. (a) 15.37 (b) 1.537

**Solution:**

Both numbers are the same, apart for the placement of decimal points. So we can find the mantissa just using the 4-digit number 1537.

To find the mantissa of 1537, just see the row labeled 15 and column labeled 3, find the number at their intersection in the log table, and add the number which is under mean difference 7 in the same row.

From the log table, 1847 + 20 = 1867. Thus;

**(a)** 15.37 = \( \scriptsize 1.537 \: \times \: 10^1 \\\scriptsize = 10^{0.1867} \: \times \: 10^1 \\ \scriptsize = 10^{0.1867 \: +\: 1} \\ \scriptsize = 10^{1.1867} \)

Hence, log 15.37 = 1.1867

**(b)** 1.537 = \( \scriptsize 1.537 \: \times \: 10^0 \\\scriptsize = 10^{0.1867} \: \times \: 10^0 \\ \scriptsize = 10^{0.1867 \: +\: 0} \\ \scriptsize = 10^{0.1867} \)

Hence, log 1.537 = 0.1867

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