Question

Log2(2-x)-log2(2x+6)=log2(-2x)-1​

Answer 1

Ответ: x₁=-2       x₂=-0,5.

Объяснение:

log₂(2-x)-log₂(2x+6)=log₂(-2x)-1

ОДЗ: 2-x>0    x<2    2x+6>0    2x>-6     x>-3    -2x>0       x<0    x∈(-3;0).

log₂((2-x)/(2x-6))=log₂(-2x)-log₂2

log₂((2-x)/(2x-6))=log₂(-2x/2)

log₂((2-x)/(2x-6))=log₂(-x)

(2-x)/(2x+6)=-x

2-x=-x*(2x+6)

2-x=-2x²-6x

2x²+5x+2=0     D=9     √D=3

x₁=-2 ∈ОДЗ      x₂=-0,5 ∈ОДЗ.

Answer 2

The given equation is:

log2(2-x) - log2(2x+6) = log2(-2x) - 1

We can simplify the left-hand side of the equation using the logarithmic identity:

log a - log b = log (a/b)

Using this identity, we can rewrite the left-hand side of the equation as:

log2[(2-x)/(2x+6)] = log2(-2x) - 1

Next, we can simplify the right-hand side of the equation using the logarithmic identity:

log a^b = b log a

Using this identity, we can rewrite the right-hand side of the equation as:

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

Simplifying further, we get:

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

This equation is an identity and is true for all values of x.

Therefore, the solution to the given equation is all real numbers.