Log2(x-1) = log4(x-1)

Question

Log2(x-1) = log4(x-1)

Answer 1

Log(2, x-1) = log(4, x-1)
ОДЗ: x-1>0
log(2^2, (x-1)^2) = log(4, x-1)
log(4, (x-1)^2) = log(4. x-1)
Отсюда (x-1)^2=(x-1),
(x-1)(x-1-1)=0
То есть
1) x-1=0 => x=1. Корень не удовлетворяет ОДЗ: 1-1=0.
2) x-2=0 => x=2. Удовлетворяет ОДЗ.
Ответ: 2.

Answer 2

Log₂(x-1) = log₄(x-1)
Т.к.  log₄(x-1)=log₂²(x-1)=log₂(x-1)/2=0.5log₂(x-1),то получаем:
log₂(x-1) = log₄(x-1) = 0.5log₂(x-1)  или log₂(x-1)- 0.5log₂(x-1)=
= 0.5log₂(x-1)=0, x-1=2⁰,x-1=1,x=2