What is solution of eq log base 3(x)+2log base x (3)=3/2?

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You need to convert the variable base of the logarithm `log_x 3` into base 3, such that:

`log_3 x + 2*(1/(log_3 x)) = 3/2`

Bringing the terms to a common denominator yields:

`2log_3 x*log_3 x + 2*2 = 3*log_3 x`

Moving the terms to the left side, yields:

`2(log_3 x)^2 - 3*log_3 x + 4 = 0`

You should come up with the following substitution, such that:

`log_3 x = y`

`2y^2 - 3y + 4 = 0`

Using quadratic formula, yields:

`y_(1,2) = (3 +- sqrt(9 - 32))/4`

Since `9 - 32 = -23 => 2y^2 - 3y + 4 != 0 => 2y^2 - 3y + 4 > 0 ` for all `y = log_3 x in R.`

**Hence, evaluating the solution to the given equation yields that there exists no solutions.**

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