Key Concepts in Computer Science – Assignment 1

1. [1 mark] What is Log4(1024)? Show how you determine the result.
𝟏𝟎𝟐𝟒= 𝟒𝟓
Log4(1024) = 𝐋𝐨𝐠𝟒(𝟒𝟓)=𝟓· 𝐋𝐨𝐠𝟒(𝟒) = 5

2. [2 marks] Given 𝐋𝐨𝐠𝟔(𝒙)=𝟐 solve for x. Show how you determine the result.
𝐋𝐨𝐠𝟔(𝒙)=𝟐 ⟹ 𝟔𝟐=𝒙 ⟹𝒙=𝟑𝟔 This is derived from the definition of logarithm.

3. [3 marks] Find the tightest integer upper bound and lower bound for x if 𝒙= 𝐋𝐨𝐠𝟐(𝟐𝟐). Justify your answer.
𝟏𝟔<𝟐𝟐< 𝟑𝟐 ⟹ 𝟐𝟒< 𝟐𝟐< 𝟐𝟓 ⟹ 𝐋𝐨𝐠𝟐(𝟐𝟒)< 𝐋𝐨𝐠𝟐(𝟐𝟐)< 𝐋𝐨𝐠𝟐(𝟐𝟓) ⟹𝟒< 𝐋𝐨𝐠𝟐(𝟐𝟐)< 5 ⟹ 𝟒<𝒙 < 5

4. [2 marks] For question 3, write the upper bound and lower bound of x using interval notation.
x ∈ (4,5)

5. [2 marks] Find answer for the following floor and ceiling functions.
⌊12.3⌋ = 12
⌈-78.999⌉ = -78
⌊-0.0001⌋ = -1
⌊10001.001⌋ = 10001

6. [3 marks] Solve for x if 2×2-3x-2 = 0 using the formula for the roots of a quadratic equation.
ax2 + bx + c = 0
x = −b ± √b2 – 4ac2a
a = 2, b = -3, c = -2
x = −(−3) ± √(−3)2 – (4×2×(-2))2×2= 3 ± √9 + 164= 3 ± √254= 3 ± 54
x = 2 and x = -0.5 (note: x has two values)

7. [3 marks] Solve this inequality 12 < -2x + 4 and write it using interval notations for x.
12 < -2x + 4 ⟹ 8 < -2x ⟹ -4 > x
x ∈ (-∞, -4)

8. [4 marks] Prove the following equation: 𝐋𝐨𝐠𝐛(𝒙𝒚)=𝐋𝐨𝐠𝐛(𝒙)− 𝐋𝐨𝐠𝐛(𝒚).
Assume b > 0, b ≠ 1, x>0, y>0
(1) Let m = Logb(x) and n = Logb(y)
(2) Then x = bm and y = bn
(3) 𝑥𝑦 = 𝑏𝑚𝑏𝑛
(4) 𝑥𝑦 = bm-n
(5) Logb(𝑥𝑦) = Logb(bm-n)
(6) Logb(𝑥𝑦) = (m – n)·Logb(b)
(7) Logb(𝑥𝑦) = m – n
(8) Logb(𝑥𝑦) = Logb(x) – Logb(y)