Упр.218 ГДЗ Алимов 10-11 класс (Алгебра)

2) 3^(2y-1) + 3^(2y-2) — 3^(2y-4) = 315;
3) 5^3х + 3* 5^(3х-2) = 140;
4) 2^(х + 1) + 3 * 2^(х-1) — 5 * 2^х + 6 = 0.

1. $$7^x-7^{x-1}=6$$

   $$7^x\left(1-\frac{1}{7}\right)=6$$

   $$7^x\cdot \frac{6}{7}=6$$

   $$7^x=7$$

   $$x=1$$

2. $$3^{2y-1}+3^{2y-2}-3^{2y-4}=315$$

   $$3^{2y}\left(\frac{1}{3}+\frac{1}{9}-\frac{1}{81}\right)=315$$

   $$3^{2y}\cdot \frac{35}{81}=315$$

   $$3^{2y}=315\cdot \frac{81}{35}=729$$

   $$3^{2y}=3^6$$

   $$2y=6,\quad y=3$$

3. $$5^{3x}+3\cdot 5^{3x-2}=140$$

   $$5^{3x}\left(1+\frac{3}{25}\right)=140$$

   $$5^{3x}\cdot \frac{28}{25}=140$$

   $$5^{3x}=140\cdot \frac{25}{28}=125$$

   $$5^{3x}=5^3$$

   $$3x=3,\quad x=1$$

4. $$2^{x+1}+3\cdot 2^{x-1}-5\cdot 2^x+6=0$$

   $$2^x\left(2+\frac{3}{2}-5\right)=-6$$

   $$2^x\left(-\frac{3}{2}\right)=-6$$

   $$2^x=4$$

   $$2^x=2^2,\quad x=2$$

Ответ

$$1)~x=1;\quad 2)~y=3;\quad 3)~x=1;\quad 4)~x=2.$$