Question

Logaritmos
Me ajudem pfvrrr.

Answer 1

Resposta:

1.

a)

$x = log_{3}(27)$

$x = log_{3}( {3}^{3} )$

$x = 3$

b)

$x = log_{ \frac{1}{5} }(125)$

$x = log_{ {5}^{ - 1} }( {5}^{3} )$

$x = - log_{5}( {5}^{3} )$

$x = - 3$

c)

$x = log_{4}( \sqrt{32} )$

$x = log_{ {2}^{2} }( \sqrt{ {2}^{5} } )$

Pelas propriedades

$log_{ {a}^{ \beta } }(b) = \frac{1}{ \beta } log_{a}(b)$

$log_{a}( \sqrt[n]{b} ) = log_{a}( {b}^{ \frac{1}{n} } ) = \frac{1}{n} log_{a}(b)$

temos:

$x = \frac{1}{2} log_{2}( {2}^{ \frac{5}{2} } )$

$x = \frac{5}{4}$

d)

$x = log_{ \frac{2}{3} }( \frac{8}{27} )$

$x = log_{ \frac{2}{3} }( \frac{2}{3} ) ^{3}$

$x = 3$

e)

$x = log_{2}( {2}^{ - 3} )$

$x = - 3$

f)

$x = log_{7}( \sqrt{7} )$

$x = \frac{1}{2}$

2.

a)

$log_{x}(8) = 3$

Temos pela definição de logaritmo:

${x}^{3} = 8$

$x = \sqrt[3]{ {2}^{3} }$

$x = 2$

b)

$log_{x}( \frac{1}{16} ) = 2$

$log_{x}( {2}^{ - 4} ) = 2$

${x}^{2} = {2}^{ - 4}$

$x = \frac{1}{4}$

c)

$log_{2}(x) = 5$

$x = {2}^{5}$

$x = 32$

d)

$x = log_{9}(27)$

$x = log_{ {3}^{2} }( {3}^{3} )$

$x = \frac{3}{2}$

e)

$x = log_{ \frac{1}{2} }(32)$

$x = log_{ {2}^{ - 1} }( {2}^{5} )$

$x = - 5$

3.

a)

$x = log( \frac{5 \times {3}^{2} }{2} )$

$x = log( \frac{45}{2} )$

$x = log(45) - log(2)$

$x = log( {3}^{2} \times 5 ) - log(2)$

$x = log( {3}^{2} ) + log(5) - log(2)$

$x = 2 log(3) + log(5) - log(2)$

b)

$x = log( \frac{ \sqrt[3]{5} \times {3}^{2} }{ {2}^{3} } )$

$x = log( \frac{9 \sqrt[3]{5} }{8} )$

$x = log(9 \sqrt[3]{5} ) - log(8)$

$x = log( {3}^{2} ) + log( \sqrt[3]{5} ) - log( {2}^{3} )$

$x = 2 log(3) + \frac{1}{3} log(5) - 3 log(2)$

4.

Seja k a expressão, assim, temos:

$k = log_{x}(12 ^{ \frac{1}{3} } )$

$k = \frac{1}{3} log_{x}(3 \times 2 \times 2)$

$k = \frac{1}{3} log_{x}(3) + \frac{1}{3} log_{x}(2) + \frac{1}{3} log_{x}(2)$

$k = \frac{1}{3} (b + a + a)$

$k = \frac{2a + b}{3}$

5.

Seja k a expressão, dessa maneira:

$k = log_{a}(10 \times 10)$

$k = log_{a}(5 \times 2 \times 5 \times 2)$

$k = 2 log_{a}(5) + 2 log_{a}(2)$

$k = 2 \times 30 + 2 \times 20$

$k = 100$

6.

a)

$k = log_{x}( \frac{ {a}^{3} }{ {b}^{2} . {c}^{4} } )$

$k = log_{x}( {a}^{3} ) - (log_{x}( {b}^{2} ) + log_{x}( {c}^{4} )$

$k = 3 log_{x}(a) - 2 log_{x}(b) - 4 log_{x}(c)$

$k = 3 \times 8 - 2 \times 2 - 4 \times 1$

$k = 16$

b)

$k = log_{x}( \frac{ \sqrt[3]{ab} }{c} )$

$k = log_{x}( \sqrt[3]{ab} ) - log_{x}(c)$

$k = \frac{1}{3} log_{x}(ab) - log_{x}(c)$

$k = \frac{1}{3} log_{x}(a) + \frac{1}{3} log_{x}(b) - log_{x}(c)$

$k = \frac{1}{3} \times 8 + \frac{1}{3} \times 2 + 1$

$k = \frac{13}{3}$

7.

a)

$k = log(24)$

$k = log(3 \times {2}^{3} )$

$k = log(3) + 3 log(2)$

$k = 3x + y$

b)

$k = log(9 \sqrt{8} )$

$k = log( {3}^{2} \times {2}^{ \frac{3}{2} } )$

$k = 2 log(3) + \frac{3}{2} log(2)$

$k = 2y + \frac{3}{2} x$

8.

a)

$log_{x - 3}(9) = 2$

Pela propriedade

$log_{a}(f(x)) = \alpha \: \: se \: \: f(x) = {a}^{ \alpha }$

temos:

$(x - 3) ^{2} = 9$

$x - 3 = 3$

$x = 6$

b)

$log_{4}(2x + 10) = 2$

$2x + 10 = 16$

$2x = 6$

$x = 3$

c)

$log_{2}( log_{3}(x - 1) ) = 2$

$log_{3}(x - 1) = 4$

$x - 1 = 81$

$x = 82$

d)

$log_{x + 1}( {x}^{2} + 7) = 2$

$(x + 1) ^{2} = {x}^{2} + 7$

${x}^{2} + 7 = {x}^{2} + 2x + 1$

$7 = 2x + 1$

$x = 3$