Examine roots x in equation log base 5 (x)+ log base x (5) = 5/2?

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You need to convert the base x of the second logarithm into the base 5, such that:

`log_x 5 = 1/(log_5 x)`

Replacing `1/(log_5 x)` for `log_x 5` yields:

`(log_5 x) + 1/(log_5 x) = 5/2`

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

`log_5 x = t`

Changing the variable yields:

`t + 1/t = 5/2`

Bringing the terms to a common denominator, yields:

`2t^2 + 2 = 5t => 2t^2 - 5t + 2 = 0`

Using quadratic formula, yields:

`t_(1,2) = (5+-sqrt(25 - 16))/4 => t_(1,2) = (5+-3)/4`

`t_1 = 2 ; t_2 = 1/2`

Replacing back `log_5 x = t` yields:

`log_5 x = 2 => x_1 = 5^2 => x_1 = 25`

`log_5 x = 1/2 => x_2 = 5^(1/2) => x_2 = sqrt 5`

Since both `x_1, x_2` are positive numbers, hence, they represent valid solutions to the given equation.

**Hence, evaluating the solutions to the given equation, yields **`x = sqrt 5, x = 25.`

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