Question

(2-log2 (x+1))/log2(x-2)=1

Answer 1

Ответ:

х∈∅

Пошаговое объяснение:

$\displaystyle \frac{2-log_2(x+1)}{log_2(x-2)}=1$
_____________________
ОДЗ:
$\displaystyle \left \{ {{\left\begin{array}{ccc}x+1 > 0\\x-2 > 0\\\end{array}\right} \atop {log_2(x-2)\neq 0}} \right. < = > \left \{ {{\left\begin{array}{ccc}x > -1\\x > 2\\\end{array}\right} \atop {x-2\neq 1}} \right. < = > \left \{ {{x > 2} \atop {x\neq 3}} \right.$
_____________________
$\displaystyle 2-log_2(x+1)=log_2(x-2)$
$\displaystyle log_2(x+1)+log_2(x-2)=2$
$\displaystyle log_2((x+1)(x-2))=2$
$\displaystyle (x+1)(x-2)=4$
$\displaystyle x^2-2x+x-2-4 = 0$
$\displaystyle x^2-x-6 = 0$
$\displaystyle D = (-1)^2-4*1*(-6)=1+24 = 25=5^2$
x₁₂ = (1±5)/(2*1)
x₁ = (1+5)/(2*1) = 6/2 = 3
x₂ = (1-5)/(2*1) = -4/2 = -2
Ни одно из решений не подходит под ОДЗ ⇒ уравнение решений не имеет

Answer 2

To solve the equation:

(2 - log2(x + 1)) / log2(x - 2) = 1

We'll start by simplifying the equation:

2 - log2(x + 1) = log2(x - 2)

Next, we'll eliminate the logarithms by rewriting them in exponential form:

2^1 - 2^(log2(x + 1)) = 2^(log2(x - 2))

Simplifying further:

2 - (x + 1) = x - 2

Now, let's solve for x:

2 - x - 1 = x - 2

Rearranging the terms:

2 - 1 + 2 = x + x

3 = 2x

Dividing both sides by 2:

x = 3/2

Therefore, the solution to the equation (2 - log2(x + 1)) / log2(x - 2) = 1 is x = 3/2.