Expand log₁₀(385)

Step 1: Factorise 385. 385 ÷ 5 = 77 77 ÷ 7 = 11 11 is prime.

So 385 = 5 × 7 × 11

Step 2: Apply the product rule. log(abc) = log a + log b + log c

log₁₀(385) = log₁₀(5 × 7 × 11) = log₁₀5 + log₁₀7 + log₁₀11

Numerical check: log₁₀5 ≈ 0.6990 log₁₀7 ≈ 0.8451 log₁₀11 ≈ 1.0414 Sum ≈ 2.5855

Verification: log₁₀(385) ≈ 2.5855 ✓

Product rule (used above): log(ab) = log a + log b

Quotient rule: log(a/b) = log a − log b

Power rule: log(aⁿ) = n log a

Change of base: logₐ(b) = log(b)/log(a)

Special values: log₁₀(1) = 0 log₁₀(10) = 1 log₁₀(100) = 2 log₁₀(1000) = 3 ln(e) = 1

More examples using product rule: • log(6) = log(2×3) = log 2 + log 3 • log(12) = log(4×3) = log 4 + log 3 = 2log 2 + log 3 • log(100) = log(10²) = 2 log 10 = 2×1 = 2

Example 1: Expand log(72) 72 = 8 × 9 = 2³ × 3² log 72 = log(2³ × 3²) = 3 log 2 + 2 log 3

Example 2: Expand log(a²b/c) = log a² + log b − log c = 2 log a + log b − log c

Example 3: Expand log(√5) = log(5^(1/2)) = (1/2) log 5

Example 4: Expand log(2 × 3 × 5) = log 2 + log 3 + log 5

Key approach:

1. Factorise the number/expression.
2. Apply product rule: log(ab) = log a + log b.
3. Apply power rule: log(aⁿ) = n log a.
4. Apply quotient rule for fractions: log(a/b) = log a − log b.