Let \(a\), \(b\), \(c\), \(l\), \(m\), \(n\) be relatively prime positive integers. In this paper it is shown that the equation \(la^ x+ mb^ y= nc^ z\), has a finite number of solutions in positive integers \(x\), \(y\), \(z\), all of which can be effectively determined. The effective procedure is based on: a) \textit{G. Frey}'s method which reduces Fermat's problem to the existence of a certain kind of elliptic curve [Ann. Univ. Sarav., Ser. Math. 1, 1-40 (1986; Zbl 0586.10010)], b) Shafarevich-Coates theorem which gives an effective method to determine (up to isomorphism) all the elliptic curves over \(\mathbb{Q}\) having good reduction at all primes outside of a fixed finite set of prime numbers [\textit{J. Coates}, Acta Arith. 16, 425-435 (1970; Zbl 0221.10027)].