Antilogarithms, Mantissa & Characteristic

1. Anatomy of a Common Logarithm

When a common logarithm is calculated, it separates into two distinct parts:

log₁₀(N) = Characteristic + Mantissa

Anatomy of Logarithmic Output

1. The Characteristic (Integral Part)

Can be zero, positive, or negative.
Numbers ≥ 1: Characteristic = (Number of digits before decimal - 1).
For 542.8 (3 digits), Characteristic = 3 - 1 = 2.
Numbers < 1 (Decimals): Characteristic = -(Number of consecutive zeros after decimal + 1).
For 0.0045 (2 zeros), Characteristic = -(2 + 1) = -3 (written as Bar 3).

2. The Mantissa (Fractional Part)

Always positive and strictly between 0 and 1.
Independent of decimal position; determined solely by the sequence of significant figures.
Retrieved directly from the standard 4-figure logarithmic tables.
Example: `log(450)`, `log(45.0)`, and `log(0.045)` all share the identical mantissa **.6532**.

The Antilogarithm is the inverse operation of finding a logarithm:

If log₁₀(x) = y, then x = Antilog(y)

Ignore the characteristic when looking up the table; locate the value corresponding to the mantissa in the antilog table.
If the characteristic is positive (+c), place the decimal point after (c + 1) digits from the left.
If the characteristic is negative (Bar c), place (c - 1) zeros immediately after the decimal point before writing the significant digits.

Problem: Evaluate using logarithmic tables:

P = (48.24 × 0.0758) ÷ 3.142

Step-by-Step Solution:

Apply Logarithms on both sides:
- log(P) = log(48.24) + log(0.0758) - log(3.142)
Determine Characteristics and find Mantissas from log tables:
- For 48.24: 2 digits before decimal → Characteristic = 2 - 1 = 1. Mantissa from table for 48, row 2, mean diff 4 is .6834.
  - log(48.24) = 1.6834
- For 0.0758: 1 zero after decimal → Characteristic = -(1 + 1) = Bar 2. Mantissa for 75, row 8 is .8797.
  - log(0.0758) = Bar 2.8797 = -2 + 0.8797
- For 3.142: 1 digit before decimal → Characteristic = 1 - 1 = 0. Mantissa for 31, row 4, mean diff 2 is .4972.
  - log(3.142) = 0.4972
Add the numerator terms:
- Numerator Sum = 1.6834 + (-2 + 0.8797)
- Add mantissas: 0.6834 + 0.8797 = 1.5631
- Add characteristics: 1 + (-2) + 1 (carried) = 0
- Numerator Log = 0.5631 (or written with integer breakdown: 1.6834 - 2 + 0.8797 = -2 + 2.5631 = 0.5631)
Subtract the denominator term:
- log(P) = 0.5631 - 0.4972
- log(P) = 0.0659
Take Antilog to evaluate P:
- P = Antilog(0.0659)
- In the Antilog table, find row .06, column 5, mean difference for 9:
  - Digits = 1161 + 3 = 1164
- Characteristic is 0, so place decimal point after (0 + 1) = 1 digit:
- Final Answer: P = 1.164