Question

Log4(3x-4)-log4(5-x2)=0,5

Answer 1

На фото..................................

Answer 2

To solve the equation:

log₄(3x - 4) - log₄(5 - x²) = 0.5

We can use the properties of logarithms. The subtraction of logarithms is the same as the logarithm of the division. Therefore, we can rewrite the equation as:

log₄((3x - 4) / (5 - x²)) = 0.5

Now, recall that the logarithmic equation logₐ(b) = c can be rewritten in exponential form as aᶜ = b.

Using this property, we can rewrite the equation again as:

4^(0.5) = (3x - 4) / (5 - x²)

Now, solve for x:

2 = (3x - 4) / (5 - x²)

Cross-multiply to eliminate the denominator:

2(5 - x²) = 3x - 4

Expand the left side:

10 - 2x² = 3x - 4

Move all terms to one side of the equation:

2x² + 3x - 14 = 0

Now, we have a quadratic equation. To solve it, we can use the quadratic formula:

x = (-b ± √(b² - 4ac)) / 2a

where a = 2, b = 3, and c = -14.

x = (-(3) ± √((3)² - 4(2)(-14))) / 2(2)

x = (-3 ± √(9 + 112)) / 4

x = (-3 ± √121) / 4

x = (-3 ± 11) / 4

Now, we have two possible solutions for x:

1. x = (11 - 3) / 4 = 2
2. x = (-11 - 3) / 4 = -14 / 4 = -7/2

So, the two solutions for the equation are x = 2 and x = -7/2.