For a candidate u and a logarithm μ algebraic over ℚ(u): uμ is real exactly when μ ∈ ℚū, and uμ is never purely imaginary

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For a candidate u and a logarithm μ algebraic over ℚ(u): uμ is real exactly when μ ∈ ℚū, and uμ is never purely imaginary · Prove2Me