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In this paper, we explicitly find all solutions of the title Diophantineequation, using lower bounds for linear forms in logarithms and properties ofcontinued fractions. Further, we use a version of the Baker-Davenport reductionmethod in Diophantine approximation, due to Dujella and Peth\"o. This paperextends the previous work of \cite{Patel}.

On the Exponential Diophantine Equation $(F_{m+1}^{(k)})^x-(F_{m-1}^{(k)})^x = F_n^{(k)}$