Question

If log2x+logx2 = 10/3 = log2y+logy2 find x+y?

Answer 1

`8 + root{3}{2}`

Explanation:

Let `log_2 x = a`. Then `log_x 2 = 1/a`. Thus we have
`a+1/a = 10/3 implies a^2-10/3 a +1 = 0`
`implies (a-3)(a-1/3) = 0`

So the values for `log_2 x` and `log_2 y` (the latter obviously satisfies the same quadratic equation) belong to the set `{3,1/3}`.
Since `x ne y`, one of the two logarithms is 3 and the other must be `1/3`. Thus one of `x` or `y` must be `2^3 = 8`, while the other one must be `2^{1/3} = root{3}{2}`.