= Solution
For the general result, choose logarithms of nonzero <algebraic numbers> $\alpha_1,\ldots,\alpha_s$ and algebraic coefficients $\beta_0,\ldots,\beta_s$, and put
$$
\Lambda=\beta_0+\sum_{j=1}^s\beta_j\log\alpha_j.
$$
Let
$$
A_j=\max\{H(f_{\alpha_j}),e^{|\log\alpha_j|},10\},
$$
where $f_{\alpha_j}$ is the <minimal polynomial> and its $H$ is the <naive polynomial height>, and let
$$
B=\max\{H(f_{\beta_0}),\ldots,H(f_{\beta_s}),
\log A_1,\ldots,\log A_s\}.
$$
The <general lower bound for a linear form in logarithms> states that, if $\Lambda\ne0$, then
$$
|\Lambda|>
\exp\!\left(-C\log A_1\cdots\log A_s\log B\right),
$$
where the effective constant $C$ depends only on $s$ and the degree of the <number field> generated by all the data.
For the improved homogeneous result, take $b_1,\ldots,b_s\in\mathbb Z$ and
$$
\Lambda=\sum_{j=1}^sb_j\log\alpha_j.
$$
With the same $A_j$, after ordering the terms set
$$
B^*=\max\left\{
\frac{|b_1|}{\log A_s},\ldots,
\frac{|b_{s-1}|}{\log A_s},|b_s|,10
\right\}.
$$
The <Baker lower bound for a homogeneous linear form in logarithms> gives
$$
\Lambda\ne0
\quad\Longrightarrow\quad
|\Lambda|>
\exp\!\left(-C\log A_1\cdots\log A_s\log B^*\right).
$$
Both constants are effective. The division by $\log A_s$ in $B^*$ is the improvement that matters when $\alpha_s$ has variable height.
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