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Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^y-y^x=c^z$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x,y,z)=(c^k+1,1,k)$, k ≥ 1 or $2\le x<y\le max1.5\times {10}^{10},c$ if c ≥ 2.