10 questions · Form 4 Additional Mathematics Bab 4: Indices, Surds and Logarithms
Use the change of base formula to evaluate log₄ 64.
Prefer reading to quizzing? All 10 questions are listed below with the answer and explanation under each one.
1. Use the change of base formula to evaluate log₄ 64.
Answer: A
log₄ 64 = log₁₀ 64 / log₁₀ 4 = log₁₀ (4³) / log₁₀ 4 = 3 log₁₀ 4 / log₁₀ 4 = 3.
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. 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.
4. 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.
5. 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.
6. Solve the logarithmic equation log₂ (x + 2) + log₂ (x - 2) = 3.
Answer: A
Combine logs: log₂ [(x+2)(x-2)] = 3 => log₂ (x² - 4) = 3 => x² - 4 = 2³ = 8 => x² = 12 => x = √12 = 2√3. (Reject x = -2√3 since log domain x > 2).
7. Which of the following numbers is an example of a surd?
Answer: C
√16 = 4, √(94) = 32, and ∛27 = 3 are all rational numbers. √7 is irrational and cannot be expressed as a ratio of integers, making it a surd.
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. 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.
10. If 3^(x + 2) = 27^(x - 1), find the value of x.
Answer: A
Express base as 3: 3^(x + 2) = (3³)^(x - 1) => 3^(x + 2) = 33x - 3) => x + 2 = 3x - 3 => 2x = 5 => x = 52.