Logarithms
A logarithm is a power: logₐ x = n means aⁿ = x. Three laws turn multiplication into addition and division into subtraction.
What a logarithm is
A logarithm answers one question: to what power must the base be raised to give this number? Since 10³ = 1,000, log₁₀ 1,000 = 3; since 2⁵ = 32, log₂ 32 = 5. Logs to base 10 are common logarithms (written log x); logs to base e (about 2.718) are natural logarithms (written ln x). Only positive numbers have logarithms, and the base must be positive and not 1. For a number of n digits before the decimal point, the whole-number part of its common log (the characteristic) is n − 1; the decimal part (the mantissa) is read from log tables.
Laws of logarithms
| Law | Rule | Example |
|---|---|---|
| Product | logₐ (mn) = logₐ m + logₐ n | log 6 = log 2 + log 3 |
| Quotient | logₐ (m/n) = logₐ m − logₐ n | log 5 = log 10 − log 2 = 1 − 0.3010 = 0.6990 |
| Power | logₐ (mⁿ) = n logₐ m | log 8 = 3 log 2 = 0.9030 |
| Base itself | logₐ a = 1 | log₅ 5 = 1 |
| Log of one | logₐ 1 = 0 | log₇ 1 = 0 |
| Change of base | logₐ x = log x ÷ log a | log₄ 32 = log₂ 32 ÷ log₂ 4 = 5/2 |
| Reciprocal | logₐ x × logₓ a = 1 | log₂ 9 × log₉ 2 = 1 |
| Number of digits | If log N = c.xxx (c whole), N has c + 1 digits before the point | log N = 3.4 → N has 4 digits |
Worked example 1: simplify with the laws
Simplify log (75/16) − 2 log (5/9) + log (32/243).
- Standard method: 2 log (5/9) = log (25/81). Combine all three: log [(75/16) ÷ (25/81) × (32/243)] = log [(75/16) × (81/25) × (32/243)].
- Cancel: 75/25 = 3, 81/243 = 1/3 and 32/16 = 2, so the product is 3 × 1/3 × 2 = 2.
- Shortcut (count prime powers): 75/16 = 3 × 5² ÷ 2⁴; 81/25 = 3⁴ ÷ 5²; 32/243 = 2⁵ ÷ 3⁵. Powers of 2: −4 + 5 = 1. Powers of 3: 1 + 4 − 5 = 0. Powers of 5: 2 − 2 = 0. Result: 2¹.
- Check: multiply out exactly: (75 × 81 × 32) ÷ (16 × 25 × 243) = 194,400 ÷ 97,200 = 2. With 4-figure logs: 0.6709 + 0.5105 − 0.8805 = 0.3009, which is log 2 (0.3010) to rounding.
log 2.
Worked example 2: how many digits?
How many digits does 2^50 have? (log 2 = 0.3010)
- Standard method: log 2^50 = 50 × 0.3010 = 15.05. The characteristic is 15, so the number has 15 + 1 = 16 digits.
- Shortcut: 2^10 = 1,024, a little more than 10³. So 2^50 = (2^10)^5 is a little more than 10^15 (about 1.13 × 10^15), which has 16 digits.
- Check: 2^50 = 1,125,899,906,842,624, which has 16 digits.
16 digits.
Worked example 3: a log equation
Solve log₂ x + log₂ (x − 2) = 3.
- Standard method: product law: log₂ [x(x − 2)] = 3, so x(x − 2) = 2³ = 8.
- x² − 2x − 8 = 0, so (x − 4)(x + 2) = 0 and x = 4 or x = −2. Logs need positive numbers, so x = −2 is rejected.
- Shortcut: you need two numbers 2 apart whose product is 8: 4 and 2. So x = 4.
- Check: log₂ 4 + log₂ 2 = 2 + 1 = 3.
x = 4.
Logarithms appear mostly in SSC, state PSC and campus placement tests as one-line value questions (log₂ 64; log 72 from log 2 and log 3) and 'how many digits' problems. Bank prelims seldom ask them directly, but the laws help in growth and compound-interest sums.
2 marks: “Find the value of log₃ 243.” 5, because 3⁵ = 243.
5 marks: “State and explain the laws of logarithms with examples.” Definition with an example; product, quotient and power laws; logₐ a = 1, logₐ 1 = 0; change of base; one numerical example for each.
10 marks: “(a) State the laws of logarithms. (b) Simplify log (75/16) − 2 log (5/9) + log (32/243). (c) Find the number of digits in 2^50 (log 2 = 0.3010).” (a) Laws with examples. (b) log 2. (c) 50 × 0.3010 = 15.05, so 16 digits.
The log is the power. Product → add, quotient → subtract, power → multiply. logₐ a = 1, logₐ 1 = 0.