Example 1 · level 1
Find the value of 3 log₄ 64 − log₄ 16 + log₄ 4.
- 2
- 8
- 12
- 65,536
Show solution
- 64 = 4³, 16 = 4², 4 = 4¹.
- So the expression is 3 × 3 − 2 + 1 = 8.
Answer: 8
ABHYAS PREP: ADVANCED QUANT · LOGARITHM
Laws of logarithms
log (xy) = log x + log y, log (x/y) = log x − log y, log xⁿ = n log x, and log_b x = log x / log b. A log is just the power: log_b x = k means bᵏ = x.
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Worked examples
Example 2 · level 2
Given log 2 = 0.3010 and log 3 = 0.4771 (base 10), find log 1,440.
- 1.1489
- 1.7781
- 2.1582
- 3.1582
Show solution
- 1,440 = 2⁴ × 3² × 10.
- log 1,440 = 4 × 0.3010 + 2 × 0.4771 + 1.
- = 3.1582.
Answer: 3.1582
Example 3 · level 3
Solve for x: log₈ x + log₈ (x + 12) = 2.
- −16
- 4
- 16
- 26
Show solution
- log₈ [x(x + 12)] = 2, so x(x + 12) = 64.
- x² + 12x − 64 = 0 ⇒ (x − 4)(x + 16) = 0.
- x = 4 or x = −16; logs need x > 0, so x = 4.
Answer: 4