A) In a survey at airport, 55 travelers said that last year they had been to Spain, 53 to France and 30 had been to France and Spain. Use Venn diagram to find the number of travelers who took part in the survey.

Tunajua kuwa $45 = 9 \times 5 = 3^2 \times 5$.
Hivyo, $\log 45 = \log (3^2 \times 5)$.

Thamani ya $\log 45$ ni $\boxed{1.6532}$.

3. a) Swali hili halijasomeka kikamilifu kutokana na picha iliyokatika. Tafadhali tuma picha kamili ya swali hili.

3. b) Kutafuta seti $A'$ na $(A \cap B)'$:

Kwanza, orodhesha vipengele vya kila seti:
Seti ya ulimwengu $U = \{1, 2, 3, ..., 25\}$
Seti $A = \{\text{nambari asilia chini ya 20}\} = \{1, 2, 3, ..., 19\}$
Seti $B = \{\text{nambari shufwa chini ya 10}\} = \{2, 4, 6, 8\}$

i) $A'$ (Kikamilisho cha A):

Seti $A'$ ni $\boxed{\{20, 21, 22, 23, 24, 25\}}$.

ii) $(A \cap B)'$ (Kikamilisho cha makutano ya A na B):

Hivi ni vipengele vilivyo katika $A$ na $B$.

Hivi ni vipengele vyote vilivyo katika $U$ lakini havipo katika $A \cap B$.

Seti $(A \cap B)'$ ni $\boxed{\{1, 3, 5, 7, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25\}}$.

4. a) Kutafuta mlinganyo wa mstari ulionyooka unaopita kwenye pointi $P(-4, 5)$ na $Q(-2, 1)$:

Kwa pointi $P(-4, 5)$ na $Q(-2, 1)$:

Mlinganyo wa mstari ni $\boxed{y = -2x - 3}$.

4. b) Mstari unaopita kwenye pointi $A(2, -k)$ na $B(4, 3)$ una mteremko wa 2:

i) Thamani ya $k$:

Kwa pointi $A(2, -k)$ na $B(4, 3)$:

Thamani ya $k$ ni $\boxed{1}$.

ii) Mlinganyo wa mstari $AB$:

Tuna pointi $A(2, -1)$ na $B(4, 3)$, na mteremko $m=2$.
Tumia mlinganyo wa mstari $y - y_1 = m(x - x_1)$ na pointi $A(2, -1)$ na mteremko $m=2$.

Mlinganyo wa mstari $AB$ ni $\boxed{y = 2x - 5}$.