Question

Log1/2 (2x-1)+log1/2 (x+3)=-2

Answer 1

Log1/2(2x-1)(x+3)=-2
log1/2(2x^2+5x-3)=-2
(1/2)^-2=2x^2+5x-3
2x^2+5x-3=4
2x^2+5x-7=0
x=1
x=-3,5

Answer 2

To solve the logarithmic equation log₁/₂(2x - 1) + log₁/₂(x + 3) = -2, we can use logarithmic properties to simplify the equation.

First, let's apply the product rule of logarithms, which states that the sum of logarithms is equal to the logarithm of the product:

log₁/₂(2x - 1) + log₁/₂(x + 3) = log₁/₂[(2x - 1)(x + 3)]

Now, let's rewrite the equation:

log₁/₂[(2x - 1)(x + 3)] = -2

Next, we can convert the equation into exponential form:

₁/₂^(log₁/₂[(2x - 1)(x + 3)]) = ₁/₂^(-2)

[(2x - 1)(x + 3)] = ₁/₂^(-2)

Simplifying the right side of the equation:

[(2x - 1)(x + 3)] = 4

Now, expand the left side of the equation:

2x² + 6x - x - 3 = 4

Combine like terms:

2x² + 5x - 3 - 4 = 0

2x² + 5x - 7 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

For the equation 2x² + 5x - 7 = 0, the coefficients are: a = 2 b = 5 c = -7

Substituting these values into the quadratic formula:

x = (-5 ± √(5² - 4 * 2 * -7)) / (2 * 2)

Simplifying:

x = (-5 ± √(25 + 56)) / 4

x = (-5 ± √81) / 4

x = (-5 ± 9) / 4

Now, we have two possible solutions:

x₁ = (-5 + 9) / 4 = 4 / 4 = 1 x₂ = (-5 - 9) / 4 = -14 / 4 = -3.5

Therefore, the solutions to the logarithmic equation log₁/₂(2x - 1) + log₁/₂(x + 3) = -2 are x = 1 and x = -3.5.