Exercise 2.3 Solved Notes
A complete solved guide for finding common logarithms, characteristics, mantissa, and solving logarithmic equations using antilogarithms, completely cleaned of all watermarks and brandings.
The logarithm of any number consists of two parts:
- Characteristic: The integral part of the logarithm. It can be positive, zero, or negative. It is determined by inspection based on the decimal position.
- For a number ≥ 1, the characteristic is (D - 1), where D is the number of digits to the left of the decimal point.
- For a number < 1, the characteristic is -(Z + 1), where Z is the number of consecutive zeros immediately after the decimal point. We write negative characteristics with a bar over them (e.g. for -2).
- Mantissa: The fractional (decimal) part of the logarithm. It is always positive and is found using logarithm tables (or calculators).
Solution:
The number 5287 can be written as 5287.0.
Number of digits before the decimal point = 4.
Characteristic = 4 - 1 = 3.
Characteristic: 3
Solution:
Number of digits before the decimal point = 2.
Characteristic = 2 - 1 = 1.
Characteristic: 1
Solution:
This decimal number is less than 1.
Number of zeros between the decimal point and the first non-zero digit (5) = 1.
Characteristic = -(1 + 1) = -2 or .
Characteristic: (or -2)
Solution:
Number of digits before the decimal point = 3.
Characteristic = 3 - 1 = 2.
Characteristic: 2
Solution:
This decimal number is less than 1.
Number of zeros between the decimal point and the first non-zero digit (4) = 4.
Characteristic = -(4 + 1) = -5 or .
Characteristic: (or -5)
Solution:
Number of digits before the decimal point = 6.
Characteristic = 6 - 1 = 5.
Characteristic: 5
Solution:
1. Find Characteristic: Number has 2 digits before decimal, so characteristic = 2 - 1 = 1.
2. Find Mantissa: Look at row 43, column 0 in the log table. Mantissa = 0.6335.
3. Combine:
Solution:
1. Find Characteristic: Number has 3 digits, so characteristic = 3 - 1 = 2.
2. Find Mantissa: Look at row 57, column 9 in the log table. Mantissa = 0.7627.
3. Combine:
Solution:
1. Find Characteristic: Number has 1 digit, so characteristic = 1 - 1 = 0.
2. Find Mantissa: Look at row 19, column 8, and difference column 2 in the log table: 2967 + 4 = 2971. Mantissa = 0.2971.
3. Combine:
Solution:
1. Find Characteristic: Number is less than 1 with 1 zero after decimal, so characteristic = -(1 + 1) = -2 or .
2. Find Mantissa: Look at row 87, column 6. Mantissa = 0.9425.
3. Combine:
Solution:
1. Find Characteristic: Number is less than 1 with 1 zero after decimal, so characteristic = -(1 + 1) = -2 or .
2. Find Mantissa: Look at row 47, column 0. Mantissa = 0.6721.
3. Combine:
Solution:
1. Find Characteristic: Number is less than 1 with 3 zeros after decimal, so characteristic = -(3 + 1) = -4 or .
2. Find Mantissa: Look at row 35, column 4. Mantissa = 0.5490.
3. Combine:
Solution:
1. Find Characteristic: 3177 has 4 digits before decimal, so characteristic = 4 - 1 = 3.
2. Find Mantissa: Since digits are same as reference (3177), the mantissa remains identical = 0.5019.
Solution:
1. Find Characteristic: 31.77 has 2 digits before decimal, so characteristic = 2 - 1 = 1.
2. Find Mantissa: Same digits as reference, so mantissa = 0.5019.
Solution:
1. Find Characteristic: Decimal number is less than 1, with 1 zero after decimal, so characteristic = -(1 + 1) = -2 or .
2. Find Mantissa: Same digits as reference, so mantissa = 0.5019.
Solution:
Given: log x = 0.0065
1. Identify characteristic and mantissa:
- Characteristic = 0
- Mantissa = 0.0065
2. Take antilogarithm: x = antilog(0.0065)
Look up row .00, column 6, difference 5 in the antilog table: 1014 + 1 = 1015.
Since the characteristic is 0, place decimal after 1 digit:
Solution:
Given: log x = 1.192
1. Identify characteristic and mantissa:
- Characteristic = 1
- Mantissa = 0.192 (or .1920)
2. Take antilogarithm: x = antilog(1.192)
Look up row .19, column 2 in antilog table: value is 1556.
Since characteristic is 1, multiply by 101 (i.e. shift decimal 1 place right):
Solution:
Given a negative log value, we must make the mantissa positive by adding and subtracting 4 (next higher integer of 3):
log x = -4 + 0.566
log x = .566
1. Identify characteristic and mantissa:
- Characteristic = -4 (or )
- Mantissa = 0.566
2. Take antilogarithm: x = antilog(.566)
Look up row .56, column 6 in the antilog table: value is 3681.
Since characteristic is -4, place 3 zeros after the decimal point:
Solution:
Add and subtract 2 (next higher integer of 1) to make mantissa positive:
log x = -2 + 0.4274
log x = .4274
1. Identify characteristic and mantissa:
- Characteristic = -2 (or )
- Mantissa = 0.4274
2. Take antilogarithm: x = antilog(.4274)
Look up row .42, column 7, difference 4: 2673 + 3 = 2676.
Since characteristic is -2, place 1 zero after decimal point:
Solution:
Given: log x = 4.3561
1. Identify characteristic and mantissa:
- Characteristic = 4
- Mantissa = 0.3561
2. Take antilogarithm: x = antilog(4.3561)
Look up row .35, column 6, difference 1: 2270 + 1 = 2271.
Since characteristic is 4, multiply by 104:
Solution:
Add and subtract 3 to make mantissa positive:
log x = -3 + 0.9816
log x = .9816
1. Identify characteristic and mantissa:
- Characteristic = -3 (or )
- Mantissa = 0.9816
2. Take antilogarithm: x = antilog(.9816)
Look up row .98, column 1, difference 6: 9572 + 13 = 9585.
Since characteristic is -3, place 2 zeros after decimal point:
Enter any positive number to compute its common logarithm, characteristic, and mantissa.
Enter a logarithm value (can be negative) to compute its antilogarithm x = 10y.
Test your characteristic and mantissa knowledge.
Reason: The number is less than 1 with 4 zeros immediately after the decimal point, so the characteristic is -(4 + 1) = -5.
Reason: The next higher integer of 2.35 is 3. Adding and subtracting 3 gives -3 + (3 - 2.35) = -3 + 0.65 = .65.
Reason: The digits are identical, so mantissa is 0.5011. The number 0.00317 is less than 1 with 2 zeros after decimal, so characteristic is -(2+1) = -3. Combined log is .5011.