Упр.378 Повторение ГДЗ Мерзляк 11 класс Базовый уровень (Алгебра)

1) 3^x-2·3^(x-2)=7; 4) 4^(x/2)+2^(x-5)-2^(x-7)=262;
2) 2^(x+1)+2^(x-3)=68; 5) 2^(x-1)+2^(x-2)+2^(x-3)=3^(x-1)-3^(x-2)+3^(x-3);
3) 7^x-(1/7)^(1-x)=6; 5) 2^(2x-1)+2^(2x-3)-2^(2x-5)=2^(7-x)+2^(5-x)-2^(3-x).

1. $$3^x-2\cdot 3^{x-2}=7$$
   $$3^x-2\cdot 3^x\cdot \frac{1}{9}=7$$
   $$3^x\left(1-\frac{2}{9}\right)=7$$
   $$3^x\cdot \frac{7}{9}=7$$
   $$3^x=9=3^2$$
   $$x=2$$
2. $$2^{x+1}+2^{x-3}=68$$
   $$2^x\left(2+\frac{1}{8}\right)=68$$
   $$2^x\cdot \frac{17}{8}=68$$
   $$2^x=32=2^5$$
   $$x=5$$
3. $$7^x-\left(\frac{1}{7}\right)^{1-x}=6$$
   $$\left(\frac{1}{7}\right)^{1-x}=7^{x-1}$$
   $$7^x-7^{x-1}=6$$
   $$7^{x-1}(7-1)=6$$
   $$6\cdot 7^{x-1}=6$$
   $$7^{x-1}=1$$
   $$x-1=0$$
   $$x=1$$
4. $$4^{x/2}+2^{x-5}-2^{x-7}=262$$
   $$2^x+2^{x-5}-2^{x-7}=262$$
   $$2^x\left(1+\frac{1}{32}-\frac{1}{128}\right)=262$$
   $$2^x\cdot \frac{131}{128}=262$$
   $$2^x=256=2^8$$
   $$x=8$$
5. $$2^{x-1}+2^{x-2}+2^{x-3}=3^{x-1}-3^{x-2}+3^{x-3}$$
   $$2^x\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}\right)=3^x\left(\frac{1}{3}-\frac{1}{9}+\frac{1}{27}\right)$$
   $$2^x\cdot \frac{7}{8}=3^x\cdot \frac{7}{27}$$
   $$\left(\frac{2}{3}\right)^x=\frac{8}{27}=\left(\frac{2}{3}\right)^3$$
   $$x=3$$
6. $$2^{2x-1}+2^{2x-3}-2^{2x-5}=2^{7-x}+2^{5-x}-2^{3-x}$$
   $$2^{2x}\left(\frac{1}{2}+\frac{1}{8}-\frac{1}{32}\right)=2^{-x}\left(128+32-8\right)$$
   $$2^{2x}\cdot \frac{19}{32}=2^{-x}\cdot 152$$
   $$19\cdot 2^{2x-5}=19\cdot 2^{3-x}$$
   $$2^{2x-5}=2^{3-x}$$
   $$2x-5=3-x$$
   $$3x=8$$
   $$x=\frac{8}{3}$$

Ответ

1) $$2$$; 2) $$5$$; 3) $$1$$; 4) $$8$$; 5) $$3$$; 6) $$\frac{8}{3}$$.