Question

Log(x-3)+log(x-2)=1-Log5​

Answer 1

X=4 это точно правильно мы тот же пример записывали в классе

Answer 2

To solve the given equation, we'll use logarithmic properties and algebraic manipulations. Here's the step-by-step solution:

1. Start with the given equation: log(x - 3) + log(x - 2) = 1 - log5.

2. We can simplify the equation by combining the logarithms on the left side using the product rule of logarithms: log[(x - 3)(x - 2)] = 1 - log5.

3. Applying the quotient rule of logarithms, we can convert the right side of the equation: log[(x - 3)(x - 2)] = log(5) - log(10).

4. Using the logarithmic identity log(a) - log(b) = log(a/b), we can simplify further: log[(x - 3)(x - 2)] = log(5/10).

5. Since the bases of the logarithms on both sides are the same, we can equate the arguments inside the logarithms: (x - 3)(x - 2) = 5/10.

6. Simplifying the right side of the equation: (x - 3)(x - 2) = 1/2.

7. Expand the left side of the equation using the FOIL method: x^2 - 5x + 6 = 1/2.

8. Multiply through by 2 to eliminate the fraction: 2x^2 - 10x + 12 = 1.

9. Move all the terms to one side to obtain a quadratic equation: 2x^2 - 10x + 12 - 1 = 0.

10. Simplify: 2x^2 - 10x + 11 = 0.

11. At this point, we have a quadratic equation. To solve it, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a).

12. For our equation, a = 2, b = -10, and c = 11. Substituting these values into the quadratic formula, we get: x = (-(-10) ± √((-10)^2 - 4(2)(11))) / (2(2)).

13. Simplifying the equation: x = (10 ± √(100 - 88)) / 4.

14. Further simplification: x = (10 ± √12) / 4.

15. The square root of 12 can be simplified: √12 = √(4 × 3) = √4 × √3 = 2√3.

16. Substituting this back into the equation: x = (10 ± 2√3) / 4.

17. Simplify the expression: x = (5 ± √3) / 2.

So, the solution to the equation log(x - 3) + log(x - 2) = 1 - log5 is x = (5 ± √3) / 2.