Question

Resolva as equações exponenciais:
Por favor me ajudem!​

Answer 1

1)

a) fatore o 81, achará 3 elevado a 4, pois 3x3x3x3=81.

${3}^{x} = 81 \\ {3}^{x} = {3}^{4} \\ x = 4$

b)

${8}^{x} = 16 \\ ({2}^{3} ) {}^{x} = {2}^{4} \\ {2}^{3x} = {2}^{4} \\ 3x = 4 \\ x = \frac{4}{3}$

c)

${7}^{x } = \frac{1}{49} \\ {7}^{x} = \frac{1}{ {7}^{2} } \\ {7}^{x} = {7}^{ - 2} \\ x = - 2$

d)

${( {2}^{x} )}^{x + 1} = 64 \\ {2}^{ {x}^{2} + x } = {2}^{6} \\ {x}^{2} + x = 6 \\ {x}^{2} + x - 6 = 0 \\ ∆= {1}^{2} - 4 \times 1 \times - 6 \\ ∆=1 + 24 = 25 \\ x1 = \frac{ - 1 + 5}{2} = \frac{4}{2} = 2 \\ x2 = \frac{ - 1 - 5}{2} = \frac{ - 6}{2} = - 3$

e)

${3}^{ {x}^{2} + 2} = {27}^{x} \\ {3}^{ {x}^{2} + 2} = {3}^{3x} \\ {x}^{2} + 2 = 3x \\ {x}^{2} - 3x + 2 = 0 \\ ∆= {( - 3)}^{2} - 4 \times 1 \times 2 \\ ∆=9 - 8 = 1 \\ x1 = \frac{ - ( -3 ) + 1}{2} = \frac{3 + 1}{2} = \frac{4}{2} = 2 \\ x2 = \frac{ - ( - 3) - 1}{2} = \frac{3 - 1}{2} = \frac{2}{2} = 1$

f)

${2}^{ {2}^{x} } = 16 \\ {2}^{ {2}^{x} } = {2}^{4} \\ {2}^{x} = 4 \\ {2}^{x} = {2}^{2} \\ x = 2$

g)

$81 \times {27}^{x} = {3}^{ {x}^{2} } \\ {3}^{4} \times {3}^{3x} = {3 }^{ {x}^{2} } \\ 4 + 3x = {x}^{2} \\ {x}^{2} - 3x - 4 = 0 \\ ∆= {( - 3)}^{2} - 4 \times 1 \times - 4 \\ ∆=9 + 16 = 25 \\ x1 = \frac{ - ( - 3) + 5}{2} = \frac{3 + 5}{2} = \frac{8}{2} = 4 \\ x2 = \frac{ - ( - 3) - 5}{2} = \frac{3 - 5}{2} = \frac{ - 2}{2} = - 1$