PART 16：例題-對數方程式

解方程式 \(\frac{1}{2}\log \left( 2x+2 \right)+\log \sqrt{3x+4}=\log 2\)
SOL:
\(\log \sqrt{2x+2}+\log \sqrt{3x+4}=\log 2\)
\(\Rightarrow \log \sqrt{(2x+2)(3x+4)}=\log 2\)
\(\Rightarrow \sqrt{(2x+2)(3x+4)}=2\)
\(\Rightarrow (2x+2)(3x+4)=4\)
\(\Rightarrow 6{{x}^{2}}+14x+8-4=0\)
\(\Rightarrow 6{{x}^{2}}+14x+4=0\)
\(\Rightarrow 3{{x}^{2}}+7x+2=0\)
\(\Rightarrow \left( 3x+1 \right)\left( x+2 \right)=0\)
\(\Rightarrow x=-\frac{1}{3}\) \(x=-2\)
驗算：真數 \(>0\) ，
(1) \(2x+2>0\) \(\Rightarrow \) \(x>-1\)
(2) \(3x+4>0\) \(\Rightarrow \) \(x>-\frac{4}{3}\)
\(\Rightarrow x=-\frac{1}{3}\) 是唯一的答案