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In this paper we prove a lower bound for the linear dependence of three positive rational numbers under certain weak linear independence conditions on the coefficients of the linear forms. Let $\Lambda =b_{2}log{\alpha }_2-b_{1}log{\alpha }_1-b_{3}log{\alpha }_3\ne 0$ with $b_1,b_2,b_3$ positive integers and ${\alpha }_1,{\alpha }_2,{\alpha }_3$ positive multiplicatively independent rational numbers greater than $1$. Let ${\alpha }_{j1}={\alpha }_{j1}/{\alpha }_{j2}$ with ${\alpha }_{j1},{\alpha }_{j2}$ coprime positive integers $(j=1,2,3)$. Let ${\alpha }_j\ge \text{max}\{{\alpha }_{j1},e\}$ and assume that gcd$(b_1,b_2,b_3)=1.$ Let $$b^{\text{'}}=\left(\frac{b_2}{log{\alpha }_1}+\frac{b_1}{log{\alpha }_2}\right)\left(\frac{b_2}{log{\alpha }_3}+\frac{b_3}{log{\alpha }_2}\right)$$ and assume that $B\ge \text{max}\{10,log{b}^{\text{'}}\}.$ We prove that either $\{b_1,b_2,b_3\}$ is $\left(c_4,B\right)$-linearly dependent over $\mathbb{Z}$ (with respect to $\left(a_1,a_2,a_3\right)$) or ...