What is the solution of 3^(9x) + 27^x = 18

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The solution of `3^(9x) + 27^x = 18` has to be determined.

`3^(9x) + 27^x = 18`

=> `(3^9)^x + (3^3)^x = 18`

=> `((3^3)^3)^x + (3^3)^x = 18`

=> `((3^x)^3)^3 + (3^x)^3 = 18`

Let `3^(3x) = y`

=> `y^3 + y - 18 = 0`

The real root of this equation is:y = `(2*sqrt(547)/3^(3/2)+9)^(1/3)-1/(3*(2*sqrt(547)/3^(3/2)+9)^(1/3)) `

To determine x equate `3^(3x) = y`

=> `x = (log y)/(3*log 3)`

The root of the given equation is `x = log((2*sqrt(547)/3^(3/2)+9)^(1/3)-1/(3*(2*sqrt(547)/3^(3/2)+9)^(1/3)) )/(3*log3)`

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