Question

2log6(2+x)+log6(9-6x+x2)=2

Answer 1

ОДЗ: 2+x>0, 9-6x+x²>0
log₆(2+x)² + log₆(9-6x+x²)=2
log₆((2+x)²(9-6x+x²))=log₆(36)
(2+x)²(x-3)²=36
((2+x)(x-3))²=36
(x²-x-6)²- 6²=0
(x²-x)(x²-x-12)=0
x²-x=0 ⇒ x(x-1)=0 ⇒x=0, x=1
x²-x-12=0 
D=1+48=49 ⇒ x₁=4, x₂=-3 (но этот корень не подходит по ОДЗ)
Ответ: х=0, х=1, х=4

Answer 2

To solve the equation 2log6(2+x) + log6(9-6x+x^2) = 2, follow these steps:

Step 1: Use logarithmic properties to simplify the equation.

• Recall the logarithmic property: log_b(x) + log_b(y) = log_b(xy).
• Apply this property to combine the logarithms on the left side of the equation: log6((2+x)^2) + log6(9-6x+x^2) = 2

Step 2: Apply another logarithmic property to remove the logarithms.

• Recall the property: log_b(x^a) = a * log_b(x).
• Using this property, rewrite the equation as: log6((2+x)^2 * (9-6x+x^2)) = 2

Step 3: Convert the logarithmic equation to an exponential equation.

• Recall that log_b(x) = y can be written as b^y = x.
• Apply this property to rewrite the equation as: 6^2 = (2+x)^2 * (9-6x+x^2)

Step 4: Solve the resulting quadratic equation. Expand the equation and bring all terms to one side to get: 36 = 4 + 4x + x^2 * 9 - 6x^2 + x^3

Now, arrange the terms to get the equation in standard form: x^3 - 6x^2 + 4x + (9 - 36) = 0 x^3 - 6x^2 + 4x - 27 = 0

Step 5: Solve the cubic equation. Solving cubic equations can be challenging, but there is no general formula like the quadratic formula. One common approach is to use numerical methods or graphing calculators to find approximate solutions.

Approximate solutions for x can be found using numerical methods like the Newton-Raphson method or by plotting the cubic equation and finding the points where it crosses the x-axis.

However, since this cubic equation can get quite complicated, I will stop here for the analytical approach. If you are looking for approximate solutions, I can assist you in finding those. Otherwise, if you need exact solutions, you'll need to use numerical methods or graphing calculators.