$\text{a)}$  

Zał:

$x^2-7>0$  

$\left(x-\sqrt{7}\right)\left(x+\sqrt{7}\right)>0$  

$x\in \left(-\infty ,-\sqrt{7}\right)\cup \left(\sqrt{7},+\infty \right)$  

 

${\log }_3\left(x^2-7\right)=2$  

${\log }_3\left(x^2-7\right)={\log }_{3}9$  

$x^2-7=9\mid +7$     

$x^2=16$  

$x=4\vee x=-4$  

---

$\text{b)}$  

Zał:

$x^2-x>0$  

$x\left(x-1\right)>0$  

$x\in \left(-\infty ,0\right)\cup \left(1,+\infty \right)$  

 

${\log }_{\frac{1}{2}}\left(x^2-x\right)=-1$  

${\log }_{\frac{1}{2}}\left(x^2-x\right)={\log }_{\frac{1}{2}}2$  

$x^2-x=2\mid -2$  

$x^2-x-2=0$  

$\Delta ={\left(-1\right)}^2-4\cdot 1\cdot \left(-2\right)=1+8=9$  

$\sqrt{\Delta }=3$  

$x_1=\frac{1-3}{2}=\frac{-2}{2}=-1$  

$x_2=\frac{1+3}{2}=\frac{4}{2}=2$  

---

$\text{c)}$  

Zał:

$9x-x^2>0$  

$x^2-9x<0$

$x\left(x-9\right)<0$  

$x\in \left(0,9\right)$  

 

${\log }_2\left(9x-x^2\right)=3$  

${\log }_2\left(9x-x^2\right)={\log }_{2}8$  

$9x-x^2=8\mid -8$  

$9x-x^2-8=0$  

$-x^2+9x-8=0$  

$\Delta =9^2-4\cdot \left(-1\right)\cdot \left(-8\right)=81-32=49$  

$\sqrt{\Delta }=7$  

$x_1=\frac{-9-7}{2\cdot \left(-1\right)}=\frac{-16}{-2}=8$  

$x_2=\frac{-9+7}{2\cdot \left(-1\right)}=\frac{-2}{-2}=1$  

---

$\text{d)}$  

Zał:

$\left|x^2-1\right|>0$  

$x^2-1\ne 0$  

$x^2\ne 1$  

$x\ne 1,x\ne -1$  

 

${\log }_2\left|x^2-1\right|=3$  

${\log }_2\left|x^2-1\right|={\log }_{2}8$  

$\left|x^2-1\right|=8$  

$x^2-1=8\vee x^2-1=-8$  

$x^2=9\vee x^2=-7$  

$x=3\vee x=-3\vee x\in \varnothing$