65 exam-style questions on Laws of exponents (rational); laws of logarithms; change of base — topic 1.7 of Number and Algebra in the IB Mathematics: Analysis and Approaches syllabus — with answers and explanations. Samples below; the full set is free with an account.
Calculate the exact value of 8^(2/3).
Answer: 4
8^(2/3) means (∛8)² = 2² = 4. A rational exponent m/n means take the nth root, then raise to the power m.
Determine the value of log₂ 40 − log₂ 5.
Answer: 3
By the quotient law, log₂ 40 − log₂ 5 = log₂ (40/5) = log₂ 8 = 3, since 2³ = 8.
Given that log a = p and log b = q, express log(a²/b) in terms of p and q.
Answer: 2p − q
log(a²/b) = log a² − log b = 2 log a − log b = 2p − q, using the power and quotient laws.
State the value of log₃ (1/27).
Answer: −3
1/27 = 3⁻³, so log₃ (1/27) = −3. A reciprocal inside the logarithm gives a negative exponent.
Solve the equation 9^x = 27.
Answer: x = 3/2
Write both sides with base 3: (3²)^x = 3³, so 2x = 3 and x = 3/2.
Express 2 ln x + ln 3 − ln y as a single logarithm (x, y > 0).
Answer: ln(3x²/y)
2 ln x = ln x²; adding logs multiplies and subtracting divides, giving ln(3x²/y). Coefficients become powers, not factors.
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