Question

Dadas as equações exponenciais abaixo, determine o conjunto solução dessas equações e assinale a resposta correta:​

Answer 1

1)

${8}^{x - 4} = 32 \\ { ({2}^{3}) }^{x - 4} = {2}^{5} \\ {2}^{3x - 12} = {2}^{5}$

$3x - 12 = 5 \\ 3x = 5 + 12 \\ 3x = 17 \\ x = \frac{17}{3}$

2)

${( \frac{3}{5} )}^{x - 1} = \frac{27}{125} \\ {( \frac{3}{5} )}^{x - 1} = \frac{ {3}^{3} }{ {5}^{3} }$

${( \frac{3}{5} )}^{x - 1} = {( \frac{3}{5} )}^{3} \\ x - 1 = 3 \\ x = 3 + 1 \\ x = 4$

3)

${10}^{3x + 1} = \frac{1}{100 \: 000} \\ {10}^{3x + 1} = \frac{1}{ {10}^{5} } \\ {10}^{3x + 1} = {10}^{ - 5}$

$3x + 1 = - 5 \\ 3x = - 1 - 5 \\ 3x = - 6 \\ x = - \frac{6}{3} \\ x = - 2$

4)

${5}^{x} = \sqrt[5]{125} \\ {5}^{x} = \sqrt[5]{ {5}^{3} } \\ {5}^{x} = {5}^{ \frac{3}{5} } \\ x = \frac{3}{5}$

5)

${27}^{x} = \frac{1}{9} \\ { ({3}^{3} )}^{x} = \frac{1}{ {3}^{2} } \\ {3}^{3x} = {3}^{ - 2} \\ 3x = - 2$

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

6)

${5}^{ {x}^{2} - 4x - 5 } = 1 \\ {5}^{ {x}^{2} - 4x - 5 } = {5}^{0}$

${x}^{2} - 4x - 5 = 0 \\ \Delta = {( - 4)}^{2} - 4.1.( - 5) \\ \Delta = 16 + 20 \\ \Delta = 36$

$x = \frac{ - b ± \sqrt{\Delta}}{2.a} \\ x = \frac{ - ( - 4) ± \sqrt{36} }{2.1}$

$x = \frac{4 ± 6 }{2}$

$x' = \frac{4 + 6}{2} = \frac{10}{2} = 5 \\ x'' = \frac{4 - 6}{2} = - \frac{2}{2} = - 1$

S={–1,5}

7)

$2. {3}^{x + 1} = 162 \\ {3}^{x + 1} = \frac{162}{2} \\ {3}^{x + 1} = 81 \\ {3}^{x + 1} = {3}^{4}$

$x + 1 = 4 \\ x = 4 - 1 \\ x = 3$

8)

${7}^{3x - 4} = 1 \\ {7}^{3x - 4} = {7}^{0}$

$3x - 4 = 0 \\ 3x = 4 \\ x = \frac{4}{3}$

9)

$\frac{ {2}^{3x + 4} }{ {2}^{x + 2} } = 16 \\ {2}^{3x + 4 - (x + 2)} = {2}^{4} \\ 3x + 4 - (x + 2) = 4 \\ 3x + 4 - x - 2 = 4$

$3x - x = 4 - 4 + 2 \\ 2x = 2 \\ x = \frac{2}{2} \\ x = 1$

10)

$3. {5}^{2x - 2} = 75 \\ {5}^{2x - 2} = \frac{75}{3} \\ {5}^{2x - 2} = 25 \\ {5}^{2x - 2} = {5}^{2}$

$2x - 2 = 2 \\ 2x = 2 + 2 \\ 2x = 4 \\ x = \frac{4}{2} \\ x = 2$