Question

Log5(1-2x)+ log1/5(-x-7)=1

Answer 1

Log_5(1-2x)+ log_1/5(-x-7)=1  1/5=5⁻¹   ОДЗ  1-2х>0   x<1/2
                                                                         -x-7>0   x<-7

Log_5(1-2x)+ log_5⁻¹(-x-7)=1

Log_5(1-2x)- log_5(-x-7)=1

Log_5(1-2x) / (-x-7)=1

 (1-2x) / (-x-7)=5¹
 
(1-2x) =5* (-x-7)

  (1-2x) = (-5x-35)


   3х=34

     x=11 1/3 не подходит под ОДЗ
                    решений нет

 

Answer 2

To solve the equation log₅(1 - 2x) + log₁/₅(-x - 7) = 1, we can use the properties of logarithms to simplify and solve for x. Let's break it down step by step:

1. Combine the logarithms using the product rule: logₐ(b) + logₐ(c) = logₐ(b * c). Applying this rule to the equation, we get: log₅(1 - 2x)(-x - 7) = 1.

2. Remove the logarithms by converting the equation into exponential form. Rewrite the equation as: (1 - 2x)(-x - 7) = 5¹.

3. Simplify the equation by expanding the brackets: (-x - 7 - 2x + 4x² + 14x) = 5.

4. Rearrange the terms and combine like terms: 4x² + 13x - 7 = 5.

5. Move all terms to one side to form a quadratic equation: 4x² + 13x - 7 - 5 = 0, 4x² + 13x - 12 = 0.

6. Solve the quadratic equation using factoring, completing the square, or the quadratic formula. Unfortunately, the quadratic equation 4x² + 13x - 12 = 0 cannot be easily factored. Let's use the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a).

   Substituting the values, a = 4, b = 13, and c = -12 into the quadratic formula, we get: x = (-13 ± √(13² - 4 * 4 * -12)) / (2 * 4).

   Simplifying further: x = (-13 ± √(169 + 192)) / 8, x = (-13 ± √(361)) / 8, x = (-13 ± 19) / 8.

7. Solve for x: a) When x = (-13 + 19) / 8: x = 6 / 8, x = 3 / 4.

   b) When x = (-13 - 19) / 8: x = -32 / 8, x = -4.

Therefore, the equation log₅(1 - 2x) + log₁/₅(-x - 7) = 1 has two solutions: x = 3/4 and x = -4.