10 questions · Form 4 Additional Mathematics Bab 4: Indices, Surds and Logarithms
Express log_a (x³ / y²) in terms of log_a x and log_a y.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Express log_a (x³ / y²) in terms of log_a x and log_a y.
Answer: A
Apply Quotient Law and Power Law: log_a (x³) - log_a (y²) = 3 log_a x - 2 log_a y.
2. Express 12 + √3 with a rational denominator.
Answer: A
Multiply by conjugate (2 - √3): 1 × (2 - √3)(2 + √3)(2 - √3) = 2 - √34 - 3 = 2 - √3.
3. Solve 5^x = 20 correct to 3 decimal places.
Answer: A
Take log₁₀ on both sides: log₁₀ (5^x) = log₁₀ 20 => x log₁₀ 5 = log₁₀ 20 => x = log₁₀ 20 / log₁₀ 5 ≈ 1.301030.69897 ≈ 1.861.
4. Given log₂ y = 3 - 2 log₂ x, express y in terms of x.
Answer: A
log₂ y + 2 log₂ x = 3 => log₂ y + log₂ (x²) = 3 => log₂ (y x²) = 3 => y x² = 2³ = 8 => y = 8x².
5. Simplify √20 + √45 - √5.
Answer: A
Simplify each surd: √20 = 2√5, √45 = 3√5. Then 2√5 + 3√5 - 1√5 = (2 + 3 - 1)√5 = 4√5.
6. What is the value of log_x x - log_x 1?
Answer: A
Since log_x x = 1 and log_x 1 = 0, we have 1 - 0 = 1.
7. Evaluate log₂ 32 - log₂ 4.
Answer: A
By the Quotient Law: log₂ (324) = log₂ 8 = log₂ (2³) = 3.
8. Find the conjugate surd of (3 - √5) and simplify their product.
Answer: A
The conjugate of (3 - √5) is (3 + √5). Product: (3 - √5)(3 + √5) = 3² - (√5)² = 9 - 5 = 4.
9. Solve the equation 22x + 1) - 5(2^x) + 2 = 0 for x.
Answer: A
Rewrite as 2(2^x)² - 5(2^x) + 2 = 0. Let u = 2^x: 2u² - 5u + 2 = 0 => (2u - 1)(u - 2) = 0 => u = 12 or u = 2. Thus 2^x = 2⁻¹ => x = -1, and 2^x = 2¹ => x = 1.
10. Simplify the expression (2³ × 4⁻¹) ÷ 81/3).
Answer: A
Express all terms with base 2: (2³ × 2⁻²) ÷ (2³)1/3) = 23 - 2) ÷ 2¹ = 2¹ ÷ 2¹ = 1.