30. Calculate the value of the expression $\log ^{2} 5+\log 2 \cdot \log 50$.

(8)
First solution. In order, we have:

$$
\begin{aligned}
\log ^{2} 5+\log 2 \cdot \log 50 & =\log ^{2} 5+\log 2 \cdot(\log 25+\log 2) \\
& =\log ^{2} 5+\log 2 \cdot(2 \log 5+\log 2) \\
& =\log ^{2} 5+2 \log 5 \cdot \log 2+\log ^{2} 2 \\
& =(\log 5+\log 2)^{2}=(\log 10)^{2}=1
\end{aligned}
$$

(full solution 8 points)