Can We Solve a Log System

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[@mystychief]

first substitute a for log x and b for log y made it easier for me using method 2.

[@phill3986]

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[@paperwhite3853]

I don't know if it includes all the possible solutions, but at a first glance sqrt(54) is sqrt(2) × sqrt(27).
in other words:
2^l1 × 3^l2 = 2^ (0.5) × 3^(1.5)
and isolating terms:
2^l1 = 2^(0.5)
3^l2 = 3^(1.5)
so: l1 = 0.5 l2 = 1.5
And sure indeed l1 + l2 = 2
x = sqrt(10)
y = sqrt(1000)

[@michaeledwards2251]

I simply looked at 54 and remembered 9 x 6 is 54.
1. The root is 3 x root(3)root(2).
2. Equating the powers of the primes, 2^log(x) = root(2) giving log(x) = 1/2, and 3^log(y) = 3 root(3) giving log(y) = 3/2. Given 1/2 + 3/2 = 2, log(x) + log(y) = 2.
3. The values of x, y are root(10) and 10 root(10) respectively.