A method is given for the resolution of diophantine equations of type \(F(2^ a\cdot 3^ b)=\pm 2^ c\cdot 3^ d\), where \(F(x)\in\mathbb{Z}[x]\) has at least two distinct roots. The method is based on lower bounds for linear forms in logarithms of algebraic numbers and the LLL-lattice basis reduction algorithm. Reviewer's remark: The author should have chosen a better example as \(F(x)=4x^ 4-3x^ 3+7x^ 2-3x+8\). Namely, considering the equation \((*)\) \(F(2^ a\cdot 3^ b)=\pm 2^ c\cdot 3^ d\) modulo 3, 16, 5, 7 and 9 one can easily see that \((*)\) has no solutions.