Question

Log (x;(x^3-8x^2-19x-10))>log (x;(5-x))+log (x;(x-2))

Answer 1

log_x(x³-8x²+19x-10)>log_x((5-x)(x-2))
1)x>0,x³-8x²+19x-10>(5-x)(x-2),...x³-7x²+12x>0,x(x-3)(x-4)>0,x€(0;3)U(4;+oo).
2)х больше 0,но меньше 1,решаем такое же нер-во,но со знаком "меньше".Общих решений не будет.
Ответ:(0;3)U(4;+оо)

Answer 2

To solve the inequality $\log_x(x^3 - 8x^2 - 19x - 10) > \log_x(5 - x) + \log_x(x - 2)$, you can start by combining the logarithms on the right-hand side using logarithm properties. The sum of two logarithms with the same base is equivalent to the logarithm of their product. Therefore, you have:

$$\log_x((5 - x) \cdot (x - 2))$$

So the inequality becomes:

$$\log_x(x^3 - 8x^2 - 19x - 10) > \log_x((5 - x) \cdot (x - 2))$$

Now, to get rid of the logarithms, you can use the fact that if two expressions have the same base and their logarithms are equal, then the expressions themselves are equal. Therefore:

$$x^3 - 8x^2 - 19x - 10 > (5 - x) \cdot (x - 2)$$

Now, let's simplify the right-hand side:

$$(5 - x) \cdot (x - 2) = (5x - 5x - x^2 + 10) = (10 - x^2)$$

So, the inequality becomes:

$$x^3 - 8x^2 - 19x - 10 > 10 - x^2$$

Now, let's simplify the inequality:

$$x^3 - 8x^2 - 19x - 10 > 10 - x^2$$

Combine like terms:

$$x^3 - 7x^2 - 19x - 10 > 10$$

Now, subtract 10 from both sides:

$$x^3 - 7x^2 - 19x - 10 - 10 > 10 - 10$$

Simplify:

$$x^3 - 7x^2 - 19x - 20 > 0$$

This is a cubic inequality. To solve it, you can find the critical points where the left side equals 0, and then test the intervals created by these critical points.

1. Find the critical points by setting the left side equal to 0:

$$x^3 - 7x^2 - 19x - 20 = 0$$

You can use numerical methods or factorization to find the solutions for this cubic equation. Once you have those solutions, they will give you the critical points.

2. Test the intervals created by the critical points using test points or by analyzing the sign of the expression. Determine when the expression is greater than 0 in each interval.

This will give you the solution to the inequality $\log_x(x^3 - 8x^2 - 19x - 10) > \log_x(5 - x) + \log_x(x - 2)$.