Question

Log3(x+2)+log3(x-6)=2

Answer 1

ОДЗ
x > 6

log 3 (x + 2)(x - 6) = log 3 (9)
(x + 2)(x - 6) = 9
x^2 - 6x + 2x - 12 - 9 = 0 
x^2 - 4x - 21 = 0
D = 16 + 4*21 = 100
x1 = (4 + 10)/2 = 14/2 = 7
x2 = (4 - 10)/2 = - 6/2 = - 3  (не удовлетворяет ОДЗ)

ОТВЕТ
7

Answer 2

To solve the logarithmic equation, we'll use the properties of logarithms. The equation you provided is:

log3(x+2) + log3(x-6) = 2

We can simplify this equation using the properties of logarithms. The sum of logarithms can be combined into a single logarithm with a product:

log3((x+2)(x-6)) = 2

Now, we can rewrite the equation in exponential form:

3^2 = (x+2)(x-6)

Simplify the left side:

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

Now, we have a quadratic equation. Let's expand the right side:

9 = x^2 - 6x + 2x - 12

Combine like terms:

9 = x^2 - 4x - 12

Move all terms to one side to set the equation to zero:

x^2 - 4x - 12 - 9 = 0

x^2 - 4x - 21 = 0

Now, we need to solve for x. We can either factor this quadratic equation or use the quadratic formula. Let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / 2a

where a = 1, b = -4, and c = -21.

x = (4 ± √((-4)^2 - 4 * 1 * (-21))) / 2 * 1

x = (4 ± √(16 + 84)) / 2

x = (4 ± √100) / 2

Now, we have two possible values for x:

x = (4 + √100) / 2 = (4 + 10) / 2 = 14 / 2 = 7

x = (4 - √100) / 2 = (4 - 10) / 2 = -6 / 2 = -3

However, we need to check if both solutions satisfy the original equation because taking the logarithm of a non-positive value is not defined.

For x = 7: log3(7+2) + log3(7-6) = log3(9) + log3(1) = 2 + 0 = 2 (satisfied)

For x = -3: log3(-3+2) + log3(-3-6) = log3(-1) + log3(-9) (not defined)

Therefore, the only valid solution is x = 7.