25 Просмотров

Задание:

Решить по алгебре:

1. 4^(x+2)=128
2. 16^(x-9)=1/2
3. 32^(2x)=4(2x+3)
4. 5(x+5)=0,04
5. 2^(12-2x)=1/8
6. 25^(x)*(1/5)^2=125^(x+1)
7. 16^(5-3x)=0,125^(5x-6)
8. (1/4)^(x-3)=256^(2x)
9. 9^(3-x)=3,24*5^(3-x)
10. 2^(x+4)-2^(x)=120
11. 3^(5-x)=7^(x-5)
12. 4^(x)-3*2^x=4
13. 4^x-2^(x+1)=48

 

Ответ на задание:

1.\( 4^{x+2} = 128 \)

\( 4^{x+2} = 4^3 \)
​\( x+2 = 3 \)​
​\( x = 1 \)​

2. ​\( 16^{x-9} = \frac{1}{2} \)​

​\( 16^{x-9} = 2^{-1} \)​
​\( x-9 = -1 \)​
​\( x = 8 \)​

3. ​\( 32^{2x} = 4(2x+3) \)​

​\( 2^{5(2x)} = 2^{2(2x+3)} \)​
​\( 5(2x) = 4(2x+3) \)​
​\( 10x = 8x + 12 \)​
​\( x = 6 \)​

4. ​\( 5(x+5) = 0.04 \)​

​\( 5x + 25 = 0.04 \)​
​\( 5x = -24.96 \)​
​\( x = -4.992 \)​

5. ​\( 2^{12-2x} = \frac{1}{8} \)​

​\( 2^{12-2x} = 2^{-3} \)​
​\( 12-2x = -3 \)​
​\( x = 7.5 \)​

6. ​\( 25^x \cdot (1/5)^2 = 125^{x+1} \)​

​\( 5^{2x} \cdot 5^{-2} = 5^{3(x+1)} \)​
​\( 2x – 2 = 3x + 3 \)​
​\( x = -5 \)​

7. ​\( 16^{5-3x} = 0.125^{5x-6} \)​

​\( 2^{4(5-3x)} = 2^{-3(5x-6)} \)​
​\( 20 – 12x = -15x + 18 \)​
​\( 3x = -2 \)​
​\( x = -\frac{2}{3} \)​

8. ​\( \left(\frac{1}{4}\right)^{x-3} = 256^{2x} \)​

​\( \left(\frac{1}{2}\right)^{2(x-3)} = 2^{8x} \)​
​\( -2(x-3) = 8x \)​
​\( x = \frac{3}{5} \)​

9. ​\( 9^{3-x} = 3.24 \cdot 5^{3-x} \)​

​\( 3^{2(3-x)} = 3.24 \cdot 5^{3-x} \)​
​\( 3(3-x) = 3.24 \)​
​\( 9 – 3x = 3.24 \)​
​\( x = \frac{9 – 3.24}{-3} \)​

10. ​\( 2^{x+4} – 2^x = 120 \)​

​\( 2^x \cdot 2^4 – 2^x = 120 \)​
​\( 16 \cdot 2^x – 2^x = 120 \)​
​\( 14 \cdot 2^x = 120 \)​
​\( x = \frac{\log\left(\frac{120}{14}\right)}{\log(2)} \)​

11. ​\( 3^{5-x} = 7^{x-5} \)​

​\( 3^{5-x} = 3^{2(x-5)} \)​
​\( 5-x = 2x – 10 \)​
​\( 3x = 5 \)​
​\( x = \frac{5}{3} \)​

12. ​\( 4^x – 3 \cdot 2^x = 4 \)​

​\( 2^{2x} – 3 \cdot 2^x = 4 \)​
​\( 2^x(2^x – 3) = 4 \)​
​\( x = \frac{\log\left(\frac{7}{2}\right)}{\log(2)} \)​

13. ​\( 4^x – 2^{x+1} = 48 \)​

​\( 2^{2x} – 2 \cdot 2^x = 48 \)​
​\( 2^x(2^x – 2) = 48 \)​
​\( x = \frac{\log(7)}{\log(2)} \)​