回復 24# maymay 的帖子
單選第 1 題：
設\(p,q\in R\)且\(p>0,q>0\)，若\(log_9 p=log_{12}q=log_{16}(p+q)\)，則\(\displaystyle \frac{q}{p}\)之值介於下列哪一個區間？
(A)\(\displaystyle (1,\frac{3}{2})\) (B)\(\displaystyle (\frac{3}{2},2)\) (C)\(\displaystyle (2,\frac{5}{2})\) (D)\(\displaystyle (\frac{5}{2},3)\) (E)\(\displaystyle (3,\frac{7}{2})\)
[解答]
令 \(\log_9 p = \log_{12} q = \log_{16}(p+q)=k,\)
則 \(p=9^k,q=12^k,p+q=16^k\)
\(\displaystyle \Rightarrow p+q=16^k=\left(\frac{12^2}{9}\right)^k=\frac{q^2}{p}\)
\(\Rightarrow p^2+pq-q^2=0\)
\(\displaystyle \Rightarrow \left(\frac{q}{p}\right)^2-\left(\frac{q}{p}\right)-1=0\)
\(\displaystyle \Rightarrow \frac{q}{p}=\frac{1\pm\sqrt{5}}{2}\)
因為 \(p>0,q>0\)，所以 \(\displaystyle \frac{q}{p}=\frac{1+\sqrt{5}}{2}\Rightarrow \frac{3}{2}<\frac{q}{p}<2\)
