Упр.838 ГДЗ Колягин Ткачёва 11 класс (Алгебра)

1. $$9^{5x}-9^{5x-1}=8$$
2. $$2^x+4-2x=120$$
3. $$5^{x+1}+5^x+5^{x-1}=155$$
4. $$3^{2x-2}\cdot3^{2x-1}-2\cdot3^{2x-2}=1$$
5. $$7^x-7^{x-1}=6$$
6. $$3^x+2+3x=10$$

1. $$9^{5x}-9^{5x-1}=8$$

   $$9^{5x}\left(1-\frac{1}{9}\right)=8$$

   $$9^{5x}\cdot \frac{8}{9}=8$$

   $$9^{5x}=9$$

   $$9^{5x}=9^1 \Rightarrow 5x=1 \Rightarrow x=\frac{1}{5}$$

2. $$2^{x+4}-2^x=120$$

   $$2^x\left(2^4-1\right)=120$$

   $$2^x\cdot 15=120$$

   $$2^x=8=2^3 \Rightarrow x=3$$

3. $$5^{x+1}+5^x+5^{x-1}=155$$

   $$5^x\left(5+1+\frac{1}{5}\right)=155$$

   $$5^x\cdot \frac{31}{5}=155$$

   $$5^x=25=5^2 \Rightarrow x=2$$

4. $$3^{2x}-2\cdot 3^{2x-1}-2\cdot 3^{2x-2}=1$$

   $$3^{2x}\left(1-\frac{2}{3}-\frac{2}{9}\right)=1$$

   $$3^{2x}\cdot \frac{1}{9}=1$$

   $$3^{2x}=9=3^2 \Rightarrow 2x=2 \Rightarrow x=1$$

5. $$7^x-7^{x-1}=6$$

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

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

   $$7^x=7=7^1 \Rightarrow x=1$$

6. $$3^{x+2}+3^x=10$$

   $$3^x\left(3^2+1\right)=10$$

   $$3^x\cdot 10=10$$

   $$3^x=1=3^0 \Rightarrow x=0$$

Ответ

1) $$\frac{1}{5}$$; 2) $$3$$; 3) $$2$$; 4) $$1$$; 5) $$1$$; 6) $$0$$.