4. In $\triangle A B C$, the sides opposite to angles $A, B, C$ are denoted as $a, b, c (b \neq 1)$, respectively, and $\frac{C}{A}, \frac{\sin B}{\sin A}$ are both roots of the equation $\log _{\sqrt{b}} x=\log _{b}(4 x-4)$. Then $\triangle A B C$
A. is an isosceles triangle, but not a right triangle
B. is a right triangle, but not an isosceles triangle
C. is an isosceles right triangle
D. is neither an isosceles triangle nor a right triangle
$$
\begin{array}{l}
\log _{\sqrt{b}} x=\log _{b} x^{2}=\log _{b}(4 x-4) \Rightarrow x^{2}=4 x-4>0 \Rightarrow x^{2}-4 x+4=0 \Rightarrow x=2, \\
\text { then } \frac{C}{A}=\frac{\sin B}{\sin A}=2 \Rightarrow\left\{\begin{array}{l}
C=2 A, \\
\sin B=\sin 3 A=2 \sin A
\end{array} \Rightarrow 3 \sin A-4 \sin ^{3} A=2 \sin A\right. \\
\Rightarrow \sin A=\frac{1}{2} \Rightarrow A=30^{\circ}, B=90^{\circ} \Rightarrow \triangle A B C \text { is a right triangle. So choose } B .
\end{array}
$$