Question

3) Determine o conjunto solução de cada equação abaixo:
a) log7 (x2 + 6x + 22) = 2
b) 2.(log4 x)² – 9log4 x + 9 = 0

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Answer 1

Resposta:

Explicação passo-a-passo:

a) log7 (x^2 + 6x + 22) = 2

x^2 + 6x + 22 = 7^2

x^2 + 6x + 22 = 49

x^2 + 6x + 22 - 49 = 0

x^2 + 6x - 27 = 0

a = 1; b = 6; c = - 27

/\ = b^2 - 4ac

/\ = 6^2 - 4.1.(-27)

/\ = 36 +81

/\= 144

V/\ = 12

x = [- b +/- V/\]/2a

x = [ - 6 +/- 12]/2.1

x = [- 6+/- 12]/2

x' = (-6-12)/2 = - 18/2= - 9

x" = (-6+12)/2 = 6/2 = 3

Resp.: {9; - 3}

Prova real:

log7 [(-9)^2 + 6.(-9)+ 22) = 2

Log 7 [81 - 54 + 22] = 2

Log 7 (103 - 54)= 2

Log 7 (49) = 2 (ok)

49 = 7^2

log7 (3^2 + 6.3 + 22) = 2

log 7 (9 + 18 + 22) = 2

log7 (49)= 2 (ok)

49 = 7^2

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b) 2.(log4 x)² – 9.log4 x + 9 = 0

(Log 4 x = y)

2y² - 9y + 9 = 0

a = 2; b = - 9; c = 9

/\ = b^2 - 4ac

/\ = (-9)^2 - 4.2.9

/\ = 81 - 72

/\ = 9

y = [ - b +/- V/\]/2a

y = [ -(-9)+/- V9]/2.2

y = [9 +/- 3]/4

y ' = [9+3]/4 = 12/4 = 3

y " = [9- 3]/4 = 6/4(:2)/(:2) = 3/2

Log 4 x = y

Log 4 x = 3

x = 4^3

x = 64

Log 4 x = y

Log 4 x = 3/2

x = 4 ^ 3/2

x = \/4^3

x = V64

x = 8

Resp.: {64; 8}

Prova real:

2.(log4 x)² – 9.log4 x + 9 = 0

2.3^2 - 9.3 + 9 = 2.9 - 27 + 9 = 18-27+9 = 9-9= 0 (ok)

2. (3/2)^2 - 9.3/2 + 9

2.9/4 - 27/2 + 9

18/4 - 27/2 + 9

9/2 - 27/2 + 9

= - 18/2 + 9 = -9+9= 0 (ok)