Question

Log6 (x^2-4)=log6 (6x-8)

Answer 1

решение на фото. наконец-то получилось прикрепить

Answer 2

log6 (x^2-4)=log6 (6x-8)

x^2-4=6x-8 , x^2-4>0 и 6x-8>0

x^2-4-6x+8=0, (х-2)(x+2)>0 и 6х>8

x^2-6x+4=0, х∈(-∞;-2)∪(2;+∞) и х>4/3

x^2 - 6x + 4 = 0 , х∈(4/3;+∞)

Найдем дискриминант квадратного уравнения:

D = b2 - 4ac = (-6)^2 - 4·1·4 = 36 - 16 = 20

Так как дискриминант больше нуля то, квадратное уравнение имеет два действительных корня:

x1 =   6 - √20 /2·1  = 3 - √5 ≈ 0.7639320225002102

x2 =   6 + √20 /2·1  = 3 + √5 ≈ 5.23606797749979, но т.к  х∈(4/3;+∞) , то подходит только второй из корней 3 + √5

Answer 3

To solve the equation, we can use the property of logarithms that states that if log_a(b) = log_a(c), then b = c. Therefore, we can set the expressions inside the logarithms equal to each other and solve for x:

log6(x^2 - 4) = log6(6x - 8)

x^2 - 4 = 6x - 8

Now, let's bring all the terms to one side of the equation:

x^2 - 6x - 4 + 8 = 0

x^2 - 6x + 4 = 0

Now, to solve this quadratic equation, we can use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

Where a = 1, b = -6, and c = 4.

x = (6 ± √((-6)^2 - 4 * 1 * 4)) / 2 * 1

x = (6 ± √(36 - 16)) / 2

x = (6 ± √20) / 2

Now, we have two possible solutions for x:

1. x = (6 + √20) / 2
2. x = (6 - √20) / 2

To simplify the solutions further, we can factor out the square root of 20:

1. x = (6 + √(4 * 5)) / 2 x = (6 + 2√5) / 2 x = 3 + √5

2. x = (6 - √(4 * 5)) / 2 x = (6 - 2√5) / 2 x = 3 - √5

So the two solutions to the equation are x = 3 + √5 and x = 3 - √5.