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Standard Four-Figure Logarithm Table (Base 10)
Verified for CBSE, State Secondary & Higher Secondary Boards, ICSE, and Competitive Examinations
No.
Main Mantissa Columns
Mean Differences (1 to 9)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
No.
Main Antilog Columns
Mean Differences (1 to 9)
0
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
7
8
9
1. Product Rule: logₐ(x · y) = logₐ(x) + logₐ(y)
The logarithm of a product equals the sum of individual logarithms. Converts complex multi-digit multiplication into simple additions.
2. Quotient Rule: logₐ(x / y) = logₐ(x) − logₐ(y)
The logarithm of a quotient equals the logarithm of the numerator minus the logarithm of the denominator.
3. Power Rule: logₐ(xⁿ) = n · logₐ(x)
The exponent of the argument becomes a multiplier in front of the logarithm.
4. Root Rule: logₐ(ⁿ√x) = (1 / n) · logₐ(x)
Roots are fractional powers. Converts high-order root extraction into straightforward division.
5. Change of Base Formula: log_b(a) = log_c(a) / log_c(b)
Allows conversion between common log (base 10), natural log (ln, base e), and binary log (base 2).
Characteristic & Mantissa Decomposition
Every common logarithm consists of two parts: log₁₀(N) = Characteristic + Mantissa.
• Characteristic: Integer component reflecting order of magnitude (can be positive, zero, or negative).
• Mantissa: Non-negative decimal fraction (0 ≤ Mantissa < 1) looked up in the 4-figure table.
Textbook Bar Notation (c̄) for Fractions
When N < 1, the characteristic is negative but the mantissa remains positive. To indicate that only the characteristic is negative, a bar (macron) is placed over it:
• log₁₀(0.0234) = 2̄.3692 = -2 + 0.3692 = -1.6308
• log₁₀(0.0054) = 3̄.7324 = -3 + 0.7324 = -2.2676 Exam Rule: Never write -2.3692 when 2̄.3692 is meant, because -2.3692 equals -2 - 0.3692 = -2.3692!
Antilogarithm Decimal Placement Rule
1. If Characteristic is positive (+c): place decimal point after (c + 1) digits from left.
Example: antilog(2.3010) → significant digits 2000 → after (2+1)=3 digits → 200.0
2. If Characteristic is negative (c̄): place (c − 1) zeroes immediately after decimal point.
Example: antilog(2̄.3010) → significant digits 2000 → (2-1)=1 zero → 0.02000