Exercício 1. Calcule:
(a) log2 16 (b) log3 81 (c) log5 125 (d) log6 1296 (e) log12 1728 (f) log2 4096 (g) log28 1
(h) log5 625 (i) log2
√
2 (j) log 100 (k) log2 1024 (l) logπ π (m) log4 16 (n) log3

1
9

(o) log81 3 (p) log 1
2
8 (q) log7

1
7

(r) log125 5 (s) log 1
2
32 (t) log9

1
27
(u) log27 81
(v) log√
8
√
32
Exercício 2. Usando apenas que log 2 = 0, 30, log 3 = 0, 47 e log 5 = 0, 69, calcule:
(a) log 4 (b) log
2
5

(c) log 12 (d) log 25 (e) log √
2 (f) log 0.5 (g) log
3
2

(h) log 20
(i) log2 3 (j) log 20+log 40+log 1600 (k) log 30 (l) log 32 (m) log
10
9

(n) log 10 (o) log2 5
(p) log 30+log 90 (q) log √
5 (r) log 15 (s) log3 5 (t) log 50−log 250 (u) log
32
15
(v) log √
6
Exercício 3. Usando apenas que log 20 = 1, 30, log 30 = 1, 47 e log 60 = 1, 79, calcule:
(a) log 4 + log 5 (b) log 5 + log 6 (c) log 150 − log 5 (d) log 120 − log 6
(e) log 5 + log 12 (f) log 180 − log 9 (g) log 16 + 2 log 5 (h) log 600 − log 30
(i) 2 log 15 + log 4 (j) log 1800 − log 60 (k) log 225 + 2 log 4 (l) log 1200 − log 20
(m) log 15 + log 40 (n) log 1800 − log 30 (o) log 20 + log 45 (p) log 6000 − log 30
2. TESTES DE VESTIBULARES

Question

Exercício 1. Calcule:
(a) log2 16 (b) log3 81 (c) log5 125 (d) log6 1296 (e) log12 1728 (f) log2 4096 (g) log28 1
(h) log5 625 (i) log2
√
2 (j) log 100 (k) log2 1024 (l) logπ π (m) log4 16 (n) log3

1
9

(o) log81 3 (p) log 1
2
8 (q) log7

1
7

(r) log125 5 (s) log 1
2
32 (t) log9

1
27
(u) log27 81
(v) log√
8
√
32
Exercício 2. Usando apenas que log 2 = 0, 30, log 3 = 0, 47 e log 5 = 0, 69, calcule:
(a) log 4 (b) log 
2
5

(c) log 12 (d) log 25 (e) log √
2 (f) log 0.5 (g) log 
3
2

(h) log 20
(i) log2 3 (j) log 20+log 40+log 1600 (k) log 30 (l) log 32 (m) log 
10
9

(n) log 10 (o) log2 5
(p) log 30+log 90 (q) log √
5 (r) log 15 (s) log3 5 (t) log 50−log 250 (u) log 
32
15
(v) log √
6
Exercício 3. Usando apenas que log 20 = 1, 30, log 30 = 1, 47 e log 60 = 1, 79, calcule:
(a) log 4 + log 5 (b) log 5 + log 6 (c) log 150 − log 5 (d) log 120 − log 6
(e) log 5 + log 12 (f) log 180 − log 9 (g) log 16 + 2 log 5 (h) log 600 − log 30
(i) 2 log 15 + log 4 (j) log 1800 − log 60 (k) log 225 + 2 log 4 (l) log 1200 − log 20
(m) log 15 + log 40 (n) log 1800 − log 30 (o) log 20 + log 45 (p) log 6000 − log 30
2. TESTES DE VESTIBULARES

Accepted Answer

Vamos resolver os exercícios passo a passo.

### Exercício 1:

(a) $\log_2 16$

$$ 
16 = 2^4 \implies \log_2 16 = 4 
$$

(b) $\log_3 81$

$$ 
81 = 3^4 \implies \log_3 81 = 4 
$$

(c) $\log_5 125$

$$ 
125 = 5^3 \implies \log_5 125 = 3 
$$

(d) $\log_6 1296$

$$ 
1296 = 6^4 \implies \log_6 1296 = 4 
$$

(e) $\log_{12} 1728$

$$ 
1728 = 12^3 \implies \log_{12} 1728 = 3 
$$

(f) $\log_2 4096$

$$ 
4096 = 2^{12} \implies \log_2 4096 = 12 
$$

(g) $\log_{28} 1$

$$ 
\log_{28} 1 = 0 
$$

(h) $\log_5 625$

$$ 
625 = 5^4 \implies \log_5 625 = 4 
$$

(i) $\log_2 \sqrt{2}$

$$ 
\sqrt{2} = 2^{1/2} \implies \log_2 \sqrt{2} = \frac{1}{2} 
$$

(j) $\log 100$

$$ 
100 = 10^2 \implies \log 100 = 2 
$$

(k) $\log_2 1024$

$$ 
1024 = 2^{10} \implies \log_2 1024 = 10 
$$

(l) $\log_\pi \pi$

$$ 
\log_\pi \pi = 1 
$$

(m) $\log_4 16$

$$ 
16 = 4^2 \implies \log_4 16 = 2 
$$

(n) $\log_3 \left(\frac{1}{9}\right)$

$$ 
\frac{1}{9} = 3^{-2} \implies \log_3 \left(\frac{1}{9}\right) = -2 
$$

(o) $\log_{81} 3$

$$ 
3 = 81^{1/4} \implies \log_{81} 3 = \frac{1}{4} 
$$

(p) $\log_{1/2} 8$

$$ 
8 = \left(\frac{1}{2}\right)^{-3} \implies \log_{1/2} 8 = -3 
$$

(q) $\log_7 \left(\frac{1}{7}\right)$

$$ 
\frac{1}{7} = 7^{-1} \implies \log_7 \left(\frac{1}{7}\right) = -1 
$$

(r) $\log_{125} 5$

$$ 
5 = 125^{1/3} \implies \log_{125} 5 = \frac{1}{3} 
$$

(s) $\log_{1/2} 32$

$$ 
32 = \left(\frac{1}{2}\right)^{-5} \implies \log_{1/2} 32 = -5 
$$

(t) $\log_9 \left(\frac{1}{27}\right)$

$$ 
\frac{1}{27} = 9^{-3/2} \implies \log_9 \left(\frac{1}{27}\right) = -\frac{3}{2} 
$$

(u) $\log_{27} 81$

$$ 
81 = 27^{4/3} \implies \log_{27} 81 = \frac{4}{3} 
$$

(v) $\log_{\sqrt{8}} \sqrt{32}$

$$ 
\sqrt{32} = (\sqrt{8})^{5/3} \implies \log_{\sqrt{8}} \sqrt{32} = \frac{5}{3} 
$$

### Exercício 2:

Usando as aproximações fornecidas: $\log 2 = 0,30$, $\log 3 = 0,47$, $\log 5 = 0,69$.

(a) $\log 4$

$$ 
\log 4 = \log(2^2) = 2 \log 2 = 2 \times 0,30 = 0,60 
$$

(b) $\log \left(\frac{2}{5}\right)$

$$ 
\log \left(\frac{2}{5}\right) = \log 2 - \log 5 = 0,30 - 0,69 = -0,39 
$$

(c) $\log 12$

$$ 
\log 12 = \log(3 \times 4) = \log 3 + \log 4 = 0,47 + 0,60 = 1,07 
$$

(d) $\log 25$

$$ 
\log 25 = \log(5^2) = 2 \log 5 = 2 \times 0,69 = 1,38 
$$

(e) $\log \sqrt{2}$

$$ 
\log \sqrt{2} = \frac{1}{2} \log 2 = \frac{1}{2} \times 0,30 = 0,15 
$$

(f) $\log 0.5$

$$ 
\log 0.5 = \log \left(\frac{1}{2}\right) = -\log 2 = -0,30 
$$

(g) $\log \left(\frac{3}{2}\right)$

$$ 
\log \left(\frac{3}{2}\right) = \log 3 - \log 2 = 0,47 - 0,30 = 0,17 
$$

(h) $\log 20$

$$ 
\log 20 = \log(4 \times 5) = \log 4 + \log 5 = 0,60 + 0,69 = 1,29 
$$

(i) $\log_2 3$

$$ 
\log_2 3 = \frac{\log 3}{\log 2} = \frac{0,47}{0,30} \approx 1,567 
$$

(j) $\log 20 + \log 40 + \log 1600$

$$ 
= \log(20 \times 40 \times 1600) = \log(1280000) 
$$

(k) $\log 30$

$$ 
\log 30 = \log(3 \times 10) = \log 3 + \log 10 = 0,47 + 1 = 1,47 
$$

(l) $\log 32$

$$ 
\log 32 = \log(2^5) = 5 \log 2 = 5 \times 0,30 = 1,50 
$$

(m) $\log \left(\frac{10}{9}\right)$

$$ 
\log \left(\frac{10}{9}\right) = \log 10 - \log 9 = 1 - (2 \log 3) = 1 - 0,94 = 0,06 
$$

(n) $\log 10$

$$ 
\log 10 = 1 
$$

(o) $\log_2 5$

$$ 
\log_2 5 = \frac{\log 5}{\log 2} = \frac{0,69}{0,30} \approx 2,3 
$$

(p) $\log 30 + \log 90$

$$ 
= \log(30 \times 90) = \log(2700) 
$$

(q) $\log \sqrt{5}$

$$ 
\log \sqrt{5} = \frac{1}{2} \log 5 = \frac{1}{2} \times 0,69 = 0,345 
$$

(r) $\log 15$

$$ 
\log 15 = \log(3 \times 5) = \log 3 + \log 5 = 0,47 + 0,69 = 1,16 
$$

(s) $\log_3 5$

$$ 
\log_3 5 = \frac{\log 5}{\log 3} = \frac{0,69}{0,47} \approx 1,468 
$$

(t) $\log 50 - \log 250$

$$ 
= \log \left(\frac{50}{250}\right) = \log \left(\frac{1}{5}\right) = -\log 5 = -0,69 
$$

(u) $\log \left(\frac{32}{15}\right)$

$$ 
= \log 32 - \log 15 = 1,50 - 1,16 = 0,34 
$$

(v) $\log \sqrt{6}$

$$ 
\log \sqrt{6} = \frac{1}{2} \log 6 = \frac{1}{2} (\log 2 + \log 3) = \frac{1}{2} (0,30 + 0,47) = 0,385 
$$

### Exercício 3:

Usando as aproximações fornecidas: $\log 20 = 1,30$, $\log 30 = 1,47$ e $\log 60 = 1,79$.

(a) $\log 4 + \log 5$

$$ 
\log 4 = 0,60 \quad \text{(calculado anteriormente)} 
$$
$$ 
\log 5 = 1,30 - 0,60 = 0,70 
$$
$$ 
\log 4 + \log 5 = 0,60 + 0,70 = 1,30 
$$

(b) $\log 5 + \log 6$

$$ 
\log 6 = \log 2 + \log 3 = 0,30 + 0,47 = 0,77 
$$
$$ 
\log 5 + \log 6 = 0,70 + 0,77 = 1,47 
$$

(c) $\log 150 - \log 5$

$$ 
\log 150 = \log(3 \times 50) = \log 3 + \log 50 = 0,47 + 1,70 = 2,17 
$$
$$ 
\log 150 - \log 5 = 2,17 - 0,70 = 1,47 
$$

(d) $\log 120 - \log 6$

$$ 
\log 120 = \log(2^3 \times 3 \times 5) = 3 \log 2 + \log 3 + \log 5 = 0,90 + 0,47 + 0,70 = 2,07 
$$
$$ 
\log 120 - \log 6 = 2,07 - 0,77 = 1,30 
$$

(e) $\log 5 + \log 12$

$$ 
\log 12 = 1,07 \quad \text{(calculado anteriormente)}
$$
$$ 
\log 5 + \log 12 = 0,70 + 1,07 = 1,77 
$$

(f) $\log 180 - \log 9$

$$ 
\log 180 = \log(2^2 \times 3^2 \times 5) = 2 \log 2 + 2 \log 3 + \log 5 = 0,60 + 0,94 + 0,70 = 2,24 
$$
$$ 
\log 9 = 2 \log 3 = 0,94 
$$
$$ 
\log 180 - \log 9 = 2,24 - 0,94 = 1,30 
$$

(g) $\log 16 + 2 \log 5$

$$ 
\log 16 = 1,20 \quad \text{(calculado anteriormente)}
$$
$$ 
2 \log 5 = 2 \times 0,70 = 1,40 
$$
$$ 
\log 16 + 2 \log 5 = 1,20 + 1,40 = 2,60 
$$

(h) $\log 600 - \log 30$

$$ 
\log 600 = \log(2^3 \times 3 \times 5^2) = 3 \log 2 + \log 3 + 2 \log 5 = 0,90 + 0,47 + 1,40 = 2,77 
$$
$$ 
\log 600 - \log 30 = 2,77 - 1,47 = 1,30 
$$

(i) $2 \log 15 + \log 4$

$$ 
\log 15 = 1,16 \quad \text{(calculado anteriormente)}
$$
$$ 
2 \log 15 = 2 \times 1,16 = 2,32 
$$
$$ 
\log 4 = 0,60 
$$
$$ 
2 \log 15 + \log 4 = 2,32 + 0,60 = 2,92 
$$

(j) $\log 1800 - \log 60$

$$ 
\log 1800 = \log(2^3 \times 3^2 \times 5^2) = 3 \log 2 + 2 \log 3 + 2 \log 5 = 0,90 + 0,94 + 1,40 = 3,24 
$$
$$ 
\log 1800 - \log 60 = 3,24 - 1,79 = 1,45 
$$

(k) $\log 225 + 2 \log 4$

$$ 
\log 225 = \log(15^2) = 2 \log 15 = 2 \times 1,16 = 2,32 
$$
$$ 
2 \log 4 = 2 \times 0,60 = 1,20 
$$
$$ 
\log 225 + 2 \log 4 = 2,32 + 1,20 = 3,52 
$$

(l) $\log 1200 - \log 20$

$$ 
\log 1200 = \log(2^4 \times 3 \times 5^2) = 4 \log 2 + \log 3 + 2 \log 5 = 1,20 + 0,47 + 1,40 = 3,07 
$$
$$ 
\log 1200 - \log 20 = 3,07 - 1,30 = 1,77 
$$

(m) $\log 15 + \log 40$

$$ 
\log 40 = \log(2^3 \times 5) = 3 \log 2 + \log 5 = 0,90 + 0,70 = 1,60 
$$
$$ 
\log 15 + \log 40 = 1,16 + 1,60 = 2,76 
$$

(n) $\log 1800 - \log 30$

$$ 
\log 1800 - \log 30 = 3,24 - 1,47 = 1,77 
$$

(o) $\log 20 + \log 45$

$$ 
\log 45 = \log(3^2 \times 5) = 2 \log 3 + \log 5 = 0,94 + 0,70 = 1,64 
$$
$$ 
\log 20 + \log 45 = 1,30 + 1,64 = 2,94 
$$

(p) $\log 6000 - \log 30$

$$ 
\log 6000 = \log(2^3 \times 3 \times 5^3) = 3 \log 2 + \log 3 + 3 \log 5 = 0,90 + 0,47 + 2,10 = 3,47 
$$
$$ 
\log 6000 - \log 30 = 3,47 - 1,47 = 2,00 
$$

Esses são os cálculos para os exercícios propostos. Se precisar de mais alguma coisa, estou à disposição!

Exercício 1. Calcule: (a) log2 16 (b) log3 81 (c) log5 125 ...

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