Question

Log0,5 (x+2) - log2 (x-3)=1/2log1/^2 (-4x-8)

Answer 1

Log₀,₅(2x-3)-(1/2)*log₀,₅(2x+3)=0   ОДЗ: 2x-3>0     x>1,5    2x+3>0    x>-1,5   ⇒   x∈(1,5;+∞)log₀,₅(2x-3)-log₀,₅(2x+3)¹/²=0log₀,₅(2x-3)-log₀,₅√(2x+3)=0log₀,₅((2x-3)/√(2x+3))=0(2x-3)/√(2x+3)=0,5⁰(2x-3)/√(2x+3)=12x-3=√(2x+3)Возведем в квадрат обе части уравнения:(2x-3)²=2x+34x²-12x+9=2x+34x²-14x+6=0   |÷22x²-7x+3=0   D=25x₁=3∈ОДЗ    x₂=0,5∉ОДЗОтвет: x=3.

Answer 2

To solve the logarithmic equation, we'll use logarithmic properties to simplify the equation and then find the value of "x".

The given equation is:

log0.5(x + 2) - log2(x - 3) = 1/2 * log(1/2)(-4x - 8)

First, let's use the logarithmic property: log(a) - log(b) = log(a/b)

log0.5(x + 2) - log2(x - 3) = log(1/2)(-4x - 8)^(1/2)

Now, let's combine the logarithms on the left side:

log0.5[(x + 2)/(x - 3)] = log(1/2)(-4x - 8)^(1/2)

Since the bases of both logarithms are the same (log base 0.5), the arguments must be equal:

(x + 2)/(x - 3) = (-4x - 8)^(1/2)

Now, we'll square both sides to eliminate the square root:

[(x + 2)/(x - 3)]^2 = (-4x - 8)

Next, let's solve for x.

(x + 2)^2 = (-4x - 8)(x - 3)^2

Expand the squared terms:

(x^2 + 4x + 4) = (-4x - 8)(x^2 - 6x + 9)

Now distribute the terms on the right side:

x^2 + 4x + 4 = -4x^3 + 24x^2 - 36x - 8x^2 + 48x - 72

Simplify the equation:

x^3 + 28x^2 - 80x + 76 = 0

Unfortunately, solving this cubic equation for "x" is not straightforward and might require numerical methods or the use of a computer algebra system. It's a polynomial equation of degree three, and the solutions might be complex numbers or irrational. If you have access to a computer algebra system or numerical solver, you can input the equation and find the solutions. Otherwise, solving it manually can be quite challenging.

However, I can help you verify the solution once you find it if you provide the value of "x".