Find x if (x^1/3)^(logx x^2 +2)=2log3 27

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justaguide eNotes educator | Certified Educator

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We have to find x for (x^1/3)^(log(x) x^2 +2)=2 log(3) 27

(x^1/3)^(log(x) x^2 +2) = 2 log(3) 27

=> (x^1/3)^(log(x) x^2 + 2) = 2 log(3) 3^3

=> (x^1/3)^(log(x) x^2 + 2) = 6 log(3) 3

=> (x^1/3)^(log(x) x^2 + 2) = 6

=> (x^1/3)^[(2* log(x) x + 2) = 6

=>...

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hala718 eNotes educator | Certified Educator

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giorgiana1976 | Student

We'll apply the power property of exponentials:

(a^b)^c = a^(b*c)

 (x^1/3)^(logx x^2 +2) =  x^(logx x^2 +2)/3

But logx x^2 = 2logx x = 2

 x^(logx x^2 +2)/3 =  x^(2 +2)/3

 x^(logx x^2 +2)/3 = x^(4/3)

x^(4/3) = 2log3 27

x^(4/3) = 2log3 (3^3)

x^(4/3) = 2*3log3 3

x^(4/3) = 6

x = 6^(3/4)

The solution of the equation is x = 6^(3/4).

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