a)

$2\cdot 4^x-3\cdot 2^{2x}+1=0$  

$2\cdot {\left(2^2\right)}^x-3\cdot 2^{2x}+1=0$  

$2\cdot 2^{2x}-3\cdot 2^{2x}+1=0$  

$-1\cdot 2^{2x}+1=0$  

$1=2^{2x}$  

$2^0=2^{2x}$  

$0=2x\mid :2$  

$0=x$  

---

b)

$4\cdot 2^{2x}-6\cdot 2^{2x}+32=0$  

$-2\cdot 2^{2x}+32=0\mid :\left(-2\right)$  

$2^{2x}-16=0$  

$2^{2x}=16$  

$2^{2x}=2^4$  

$2x=4\mid :2$  

$x=2$  

---

c)

$3^{2x-10}-30\cdot 3^{x-5}+81=0$  

$3^{2\cdot \left(x-5\right)}-30\cdot 3^{x-5}+81=0$  

${\left(3^{x-5}\right)}^2-30\cdot 3^{x-5}+81=0$  

Podstawmy  $3^{x-5}=t$  

$t^2-30t+81=0$  

$\Delta ={\left(-30\right)}^2-4\cdot 1\cdot 81=900-324=576$  

$\sqrt{\Delta }=24$  

$t_1=\frac{30-24}{2}=\frac{6}{2}=3$  

$t_2=\frac{30+24}{2}=\frac{54}{2}=27$  

Zatem otrzymujemy:

$3^{x-5}=3\vee 3^{x-5}=27$  

$3^{x-5}=3^1\vee 3^{x-5}=3^3$  

$x-5=1\vee x-5=3$  

$x=6\vee x=8$  

---

d)

$5^{4x+1}-626\cdot 5^{3x}+125\cdot 5^{2x}=0$  

$5^{4x}\cdot 5^1-626\cdot 5^{3x}+125\cdot 5^{2x}=0$  

${\left(5^x\right)}^4\cdot 5-626\cdot {\left(5^x\right)}^3+125\cdot {\left(5^x\right)}^2=0$  

Podstawmy  $5^x=t$  

$5t^4-626t^3+125t^2=0$  

$t^2\left(5t^2-626t+125\right)=0$  

$t^2=0\vee 5t^2-626t+125=0$   

$\Delta ={\left(-626\right)}^2-4\cdot 5\cdot 125=391876-2500=389376$  

$\sqrt{\Delta }=624$  

$t_1=\frac{626-624}{2\cdot 5}=\frac{2}{10}=\frac{1}{5}$  

$t_2=\frac{626+624}{2\cdot 5}=\frac{1250}{10}=125$  

$t=0\vee t=\frac{1}{5}\vee t=125$  

Zatem otrzymujemy:

$5^x=0\vee 5^x=\frac{1}{5}\vee 5^x=125$  

$\text{sprzecznosˊcˊ}\vee 5^x=5^{-1}\vee 5^x=5^3$  

$x=-1\vee x=3$