3^(x+23) = 1/9
3^(x+23) = 1/ 3^2
3^(x+23) = 3^(-2)
x+23=-2
x= -2 - 23
x= - 25
4=2^( (23x-1) /(23x-2) )
2^2 = 2^( (23x-1)/(23x-2) )
2 = (23x - 1) /(23x-2)
23x-2≠ 0 ⇒ x≠2/23
2(23x-2) = 23x-1
46x-4 =23x-1
46x-23x=-1+4
23x = 3
x=3/23
(67/5) ^(23x-10) = (1/24)^ (23x-10)
замена переменной : 23x-10=t
(67/5)^t = (1/24)^t
67^t / 5^t = 1^t / 24^t
67^t * 24^t = 5^t * 1^t
(67*24)^t = (5*1)^t
1608 ^t = 5^t |: 5^t
1608^t / 5^t = 1
(1608/5)^t = (1608/5)^0
t=0
подставим:
23x - 10=0
23х=10
х=10/23
26^(2x/23 -1) + 26^(2x/23) = 27
26^(2x/23) / 26^1 + 26^(2x/23)=27
26^(2x/23) * (1/26 +1 ) = 27
26^(2x/23) * 27/26 = 27
26^(2x/23) = 27 : 27/26
26^(2x/23) = 27/1 * 26/27
26^(2x/23) = 26
26^(2x/23) = 26^1
2x/23 = 1
2x= 1*23
x= 23/2
x= 11,5
24^(x-1) = 1/![\sqrt[23]{24}](https://tex.z-dn.net/?f=+%5Csqrt%5B23%5D%7B24%7D+ "\sqrt[23]{24}")
24^(x-1) = 1/ 24^(1/23)
24^(x-1) = 24^ ( - 1/23)
x-1= - 1/23
x= - 1/23 + 1
x= 22/23
3 * 4^x + 3*4^(x+1) + 4^(x+2) = 62
3 *4^x +3 * 4^x * 4 + 4^x * 4^2 = 62
замена переменной 4^x=t
3t + 12t + 16t = 62
31t=62
t=62/31
t= 2
4^x= 2
(2^2)^x = 2
2^2x=2^1
2x=1
x=1/2
x=0.5
3^2x - 2*3^x - 3 = 0
(3^x)^2 - 2*3^x -3 = 0
3^x = t
t^2 - 2*t - 3 = 0
D= (-2)^2 - 4*1*(-3) = 4 + 12 = 16 =4^2
D>0 - два корня уравнения
t1= (2-4)/(2*1) = -2/2 = -1
t2= (2+4)/(2*1) = 6/2=3
подставим:
3^x = - 1 нет комплексных корней
3^x =3
3^x =3^1
x=1