NCERT Solutions for Class 11 Maths Chapter 12 Introduction to Three Dimensional Geometry

March 23, 202339 Insightful Publications

**Log Table:** In Mathematics, the Logarithm is the inverse operation to exponentiation. It means the Logarithm of a number is the exponent to which the base must be raised to produce that number. In simple cases, the Logarithm counts repeated multiplication. To find the value of a Logarithmic Function, students have to use the log table. The Logarithm Table is highly effective in finding the value of the logarithmic function.

At Embibe, we have provided the free PDF of the log table on this page along with the table definition. Moreover, we have explained in detail with illustrations how to use the Logarithm Table and the procedure to use the Antilogarithm Table. Read further to find everything about the Log Table.

Before we provide you with the Logarithm Table from which students will get all the values, let us understand the concept of the Logarithmic Function. Logarithmic Functions are the inverses of Exponential Functions. A Log Function is defined as follows:

For x, b > 0, and b≠1,f(x) = log_{b }x, if x = b^{y} |

Here, f(x) is the **Logarithm** Function to the base ‘b’. The most common bases used in Log Functions are base e and base 10.

**Common Logarithm [f(x) = log**_{10}**x]**: The Logarithm to base 10 (that is b = 10) is called the Common Logarithm and has many applications in Science and Engineering.

**Natural Logarithm** **[f(x) = log**_{e}**x]**: The Natural Logarithm has the number e (**≈ 2.718**) as its base. Its use is widespread in Maths and Physics because of its simpler derivative.

**Binary Logarithm** **[f(x) = log**_{2}**x]**: The Binary Logarithm uses base 2 (that is b = 2) and is commonly used in Computer Science.

The integral part of a Common Logarithm is called the **Characteristic** and the fractional part is called the **Mantissa**.

**Note:** The Mantissa part of the log of a number is always kept positive.

Thus, the log of any number *N* will be in the form:

logN = (Characteristic)⋅(Mantissa) N = antilog (Mantissa) × 10^{Characteristic} |

Here we have provided the log values table for the base 10.

How to Use Log Table?: Step-By-Step Process with Example

To find the log value of a number using the log table, students must understand the process of reading the log table. We have provided a step-by-step process to find the values using an example:

**Step 1:** Identify the table. For different bases, a different log table is used. The table provided above is for base 10. So, you can find the log value of a number to the base 10 only. To find Natural Logarithms or Binary Logarithms, students will have to use a different table.

**Step 2:** Find out the integer and decimal parts of the given number. Suppose we want to find the log value of n = 18.25. So, first of all, we separate the integer and decimal.

Integer Part: 18

Decimal Part: 25

**Step 3:** Come to the common log table and look for the cell value at the following intersections:

The row labelled with the first two digits of n

Column header with the third digit of n

⇒ In this example, log10(18.25) → row 18, column 2 → cell value 2601. So, the value obtained is 2601.

**Step 4:** Always use the Common Logarithm table with a mean difference. Now again go to row number 18 and column number 5 (fourth digit of n) in the mean difference table.

⇒ In this example, log_{10}(18.25) → row 18, mean difference column 5 → cell value 12. Write down the corresponding value which is 12.

**Step 5:** Add both the values obtained in step 3 and step 4.

That is 2601 + 12 = 2613.

**Step 6:** Find the characteristic part. By trial and error, find the integer value of *p* such that a^{p} < n and a^{p+1} > n. Here a is the base and p is the characteristic part. For common (base 10) logs, just count the number of digits left of the decimal and subtract one.

So, characteristic part = (number of digits to the left of the decimal – 1).

In this example, characteristic = 2 – 1

= 1

**Step 7:** Combine both the characteristic and the Mantissa part and students will get the final value which is 1.2613.

So, log_{10} (18.25) = 1.2613

Here we have provided all the important laws and properties related to Logarithms.

**Theorem 1**: The Logarithm of the product of two numbers say a and b, is equal to the sum of the Logarithm of the two numbers. The base should be the same for both numbers.

This theorem is also known as the **Product Rule for Logarithms**.

**Theorem 2**: The division of the two numbers is the Antilog of the difference of the Logarithm of the two numbers.

In other words, the Logarithm of the division of two numbers, say a and b, is equal to the difference of the Logarithm of the two numbers. The base should be the same for both numbers.

This theorem is also called the **Quotient Rule for Logarithms**.

**Theorem 3**: The Logarithm of a number to any other base can be determined by the Logarithm of the same number to any given base.

**Theorem 4**: The Logarithm of a number raised to a power is equal to the index of the power multiplied by the Logarithm of the number. The base is the same in both.

This theorem is also known as the **Power Rule for Logarithms**.

So, these are the 4 Logarithm properties. Students will be able to rewrite a Logarithmic expression using the power rule, product rule, or quotient rule for Logarithms.

Find the common log tables 1 to 10 below:

Common Logarithm to a Number (log_{10}x) | Log Values |

Log 1 | 0 |

Log 2 | 0.3010 |

Log 3 | 0.4771 |

Log 4 | 0.6020 |

Log 5 | 0.6989 |

Log 6 | 0.7781 |

Log 7 | 0.8450 |

Log 8 | 0.9030 |

Log 9 | 0.9542 |

Log 10 | 1 |

Natural Logarithm Table for 1 to 10

Find the natural logarithmic values table for 1 to 10 below:

Natural Logarithm to a Number (log_{e}x) | Log Values |

ln (1) | 0 |

ln (2) | 0.693147 |

ln (3) | 1.098612 |

ln (4) | 1.386294 |

ln (5) | 1.609438 |

ln (6) | 1.791759 |

ln (7) | 1.94591 |

ln (8) | 2.079442 |

ln (9) | 2.197225 |

ln (10) | 2.302585 |

*Get Algebra formulas for Classes 8 to 11 below:*

Algebra Formulas for Class 8 | Algebra Formulas for Class 9 |

Algebra Formulas for Class 10 | Algebra Formulas for Class 11 |

Here we have provided some of the sample questions and answers on log tables 1 to 100:

Question 1: Find the value of log_{10} 8.675 |

Solution: The value can be obtained in the following steps:Step 1: Integer = 8 and Decimal = 675Step 2: Check row number 86 (first two digits of the given number) and column number 7 (third digit of the given number). So, the value obtained is 9380.Step 3: Check the mean difference value for row number 86 and the mean difference in column 5. The value corresponding to the row and column is 3. Step 4: Add the values obtained in steps 2 and 3, we get 9383. This is the mantissa part. Step 5: Since the number of digits to the left side of the decimal part is 1. So the characteristic part = (Number of digits to the left of the decimal – 1) = 0 Step 6: Combine the characteristic and the mantissa part. So, it becomes 0.9383. Therefore, the value of log_{10} 8.675 is 0.9383. |

Question 2: Find the value of log (45.67) using the log table. |

Solution: 45.67 = 4.567 × 10^{1}So, log (45.67) = log (4.567 x 10) = log (4.567) + log (10) [Using the logarithm property: log (a.b) = log a + log b] = log (4.567) + 1 [∵ log (10) = 1, for common log] Now, let’s find the value of the log (4.567). Look up a standard log table. Go to row number 45 (first two digits of n) and column number 6 (third digit of n). Note down the corresponding value which is 0.6590. Now again go to row number 45 and column number 7 (fourth digit of n) in the mean difference table. Note down the corresponding value which is 7. Now add the two values. We get: 0.6590 + 7 = 0.6597 This is the Mantissa part. Since, we are using a common log table, characteristic = (Number of digits to the left of decimal – 1) = (1 – 1) = 0 ∴ Characteristic = 0 Now, combine both the parts, we get log (4.567) = 0.6597 log (45.67) = log (4.567) + 1 = 0.6597 + 1 1.6597 Therefore, the value of log (45.67) is 1.6597 |

Question 3: Use a log table to evaluate the following logarithmic function: |

Solution: We can solve the value of N in 4 steps:Step 1: Convert the expression for N into simple logs.Step 2: Evaluate those logs using a log table.Step 3: Determine the value of log N.Step 4: Calculate the value of N using the antilog of log N.So, logN = log(647⋅32×0.00000147 / 8.473×64) = log(647⋅32) + log(0.00000147) − log(8.473) − log(64) [Using the product rule for logarithms] Using the log table, we find the following values: 647.32 = 6.4732 × 10 ^{2}⇒ log(647.32 ) = 2 + log(6.4732) = 2 + 0.8111 = 2.8111 0.00000147 = 1.47×10 ^{-6}⇒ log(0.00000147) = −6 + log(1.47) = -6 + 0.1673 8.473 = 8.473×10 ^{0} ⇒ log(8.473) = 0.9280 64 = 6.4 × 10 ^{1} ⇒ log(64) = 1 + log(6.4) = 1.8062 ∴ logN = 2.8111 + (-6 + 0.1673) – 0.9280 – 1.8062 = –5.7558 = –5 – 0.7558 = (–5 – 1) + 1 – 0.7558 [∵ Mantissa cannot be negative, we have added and subtarcted 1 to make it positive] = –6 + 0.2442 Now, we have the characteristic = -6 and the Mantissa = 0.2442. To find the value of N, we use the formula: N = antilog (Mantissa) × 10^{Characteristic} = antilog (0.2442) x 10 ^{-6}= 1.7547 x 10 ^{-6} [Using the antilog table]⇒ N = 1.7547 × 10 ^{-6} = 0.0000017547 Hence, the value of N is 0.0000017547. You can verify the value using a calculator. |

Here we have provided some of the practice questions on the mathematical log table for practice:

Q1: Find logarithm values of the following to the base 10 using the log table: (i) 7.653 (ii) 14.25 (iii) 9.281 Q2: If 3 + log _{10} x = 2 log_{10} y, find x in terms of y.Q3: Prove that, 7 log (10/9) + 3 log (81/80) = 2log (25/24) + log 2. Q4: If log:_{10} 2 = 0.30103, log_{10} 3 = 0.47712 and log_{10} 7 = 0.84510, find the values of(i) log_{10} 45(ii) log _{10} 105Q5: If log 9 = b and log 5 = c, evaluate log 75 in terms of b and c. |

The Antilog table is otherwise known as the Anti-Logarithm table. Antilog of a number is the inverse method to find the log of the existing or same number. For example, if a is the log of a number b with base x, then we can simply say b is the antilog of a with base x.

**If log _{x} b = a Then, b = antilog a**

The base of both Logarithm and Antilog is 2.7183. If the base of the Logarithm and Antilogarithm is 10, they should be multiplied by 2.303 to get a Natural Logarithm and Antilog.

*Check other important Maths articles:*

**Q.1: How do you calculate Logs?Ans:** The power to which a base of 10 must be raised to obtain a number is called the Common Logarithm (Log) of the number. To calculate the log values, we use the Mathematics Logarithm Table.

**Q.2: How do you find the Log of a number without a calculator?****Ans:** You can find the Log of a number without a calculator using the Logarithmic and Mathematical tables provided on this page.

**Q.3: What is the value of log 1 to 10?****Ans:** Log 1 to the base 10 is equal to 0.

**Q.4: How do you read a Log Table?****Ans: **Take the first 2 digits of the number irrespective of the decimal and look for the row with that number. Next, look for the column number corresponding to the third digit of the number. You may also need to look into the mean difference table to get the final value. The step-by-step process to read a Log table is provided on this page.

**Q.5: What is the value of log 0?****Ans:** Log 0 is undefined. It is not a real number because you can never get zero by raising anything to the power of anything else.

**Q.6: What is the value of e in the Natural Logarithm?Ans:** e is an irrational and transcendental number approximately equal to 2.7182.