Question

1 Resolva a equação:
${2}^{ x + 4} = 64$
2 Calcule o valor de x na equação:
$16 {}^{x} = \frac{1}{ {4}^{x} }$
3
$(2|5) {}^{3x} = 25|4$
4
$2 {}^{x} + {2}^{x + 1} + {2}^{x + 2} + {2}^{x + 3} = \frac{15}{2}$
5
${2}^{x - 3} + {2}^{x - 1} + {2}^{x} = 52$


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Answer 1

$1)2^{x + 4} = 64 \\ 2^{x + 4} = {2}^{6} \\ x + 4 = 6 \\ x = 6 - 4 \\ x = 2$

$2) {16}^{x} = \frac{1}{ {4}^{x} } \\ {2}^{4x} = {2}^{ - 2x} \\ 4x = - 2x \\ 4x + 2x = 0 \\ 6x = 0 \\ x = 0$

$3)( \frac{2}{5} {)}^{3x} = \frac{25}{4} \\ ( \frac{2}{5} {)}^{3x} = ( \frac{2}{5} {)}^{ - 2} \\ 3x = - 2 \\ x = - \frac{2}{3}$

$4)4 + {2}^{x + 1} + 2 ^{x + 2} + {2}^{x + 3} = \frac{15}{2} \\ {2}^{x + 1} + {2}^{x + 2} + {2}^{x + 3} = \frac{15}{2} - 4 \\ (1 + 2 + {2}^{2} ) \times {2}^{x + 1} = \frac{7}{2} \\ 7 \times {2}^{x + 1} = \frac{7}{2} \\ {2}^{x + 1} = \frac{1}{2} \\ {2}^{x + 1} = {2}^{ - 1} \\ x + 1 = - 1 \\ x = - 1 - 1 \\ x = - 2$

$5)(1 + {2}^{2} + {2}^{3} ) \times {2}^{x - 3} = 52 \\ (1 + 4 + 8) \times {2}^{x - 3} = 52 \\ 13 \times {2}^{x - 3} = 52 \\ {2}^{x - 3} = 4 \\ {2}^{x - 3} = {2}^{2} \\ x - 3 = 2 \\ x = 2 + 3 \\ x = 5$