Example 1: simplifying a difference of two logarithms

1. Simplify log₂ 24 − log₂ 3.
2. log₂ 24 − log₂ 3 = log₂ (24 / 3) = log₂ 8 = 3.

Example 2: adding two logarithms

• Find log 4 + log 25 without a calculator, where the base is 10.
• Solution log 4 + log 25 = log (4 × 25) = log 100 = 2, because 10² = 100.

Example 3: expanding a logarithmic expression

• Write log_a (x³ y / z²) as a sum or difference of logarithms, where x, y and z are positive numbers.
• Solution log_a (x³ y / z²) = log_a (x³) + log_a y − log_a (z²) = 3 log_a x + log_a y − 2 log_a z.

Example 4: the change of base rule

• Find log₄ 8 without a calculator.
• Solution Change the base to 2: log₄ 8 = log₂ 8 / log₂ 4 = 3 / 2.

Example 5: an equation with two logarithms of the same base

• Solve log₂ x + log₂ (x − 2) = 3.
• Solution By the product law, log₂ (x(x − 2)) = 3, so x(x − 2) = 2³ = 8. This gives x² − 2x − 8 = 0, and factorising (x − 4)(x + 2) = 0 gives x = 4 or x = −2.
• The domain requires x > 0 and x − 2 > 0, that is x > 2. So we reject −2 and keep x = 4.
• Check log₂ 4 + log₂ 2 = 2 + 1 = 3. The solution is correct.

Example 6: an equation with a difference of two logarithms

• Solve log₃ (x + 1) − log₃ (x − 1) = 1.
• Solution By the quotient law, log₃ ((x + 1) / (x − 1)) = 1, so (x + 1) / (x − 1) = 3. Multiplying by (x − 1) gives x + 1 = 3x − 3, so 2x = 4 and x = 2.