Question

Log 5 (2/x +2) - log 5 (x+3) меньше или ровно log 5 (x+6/x^2)

Answer 1

Ответ:

Пошаговое объяснение:

$log_5(\frac{2}{x+2} )-log_5(x+3)$

$log_5\frac{2}{(x+2)(x+3)}$

Функция $y = log_5(x)$ - возрастающая, поэтому логарифмы можно убрать, и знак неравенства останется тот же.

$\frac{2}{(x+2)(x+3)}$

$\frac{2}{(x+2)(x+3)} -\frac{x+6}{x^2}$

$\frac{2x^2-(x+2)(x+3)(x+6)}{x^2(x+2)(x+3)}$

$\frac{2x^2-(x^3+5x^2+6x+6x^2+30x+36)}{x^2(x+2)(x+3)}$

$\frac{-x^3-9x^2-36x-36}{x^2(x+2)(x+3)}$

Меняем все знаки в числителе, при этом меняется знак неравенства

$\frac{x^3+9x^2+36x+36}{x^2(x+2)(x+3)} >=0$

Числитель имеет единственный корень x1 ≈ -1,43. Можно заменить:

$\frac{x+1,43}{x^2(x+2)(x+3)} >=0$

По методу интервалов получаем решение:

x ∈ (-3; -2) U (-1,43; 0) U (0; +oo)

Answer 2

To compare the expressions involving logarithms, we need to use the properties of logarithms. The properties we will use are:

1. log_b(a) - log_b(c) = log_b(a/c)
2. log_b(a) ≤ log_b(c) if and only if a ≤ c (for positive values of a, b, and c)

Let's start by simplifying the given expression step by step:

Given expression: log₅(2/x + 2) - log₅(x + 3) ≤ log₅((x + 6)/x²)

Step 1: Apply property 1 log₅((2/x + 2)/(x + 3)) ≤ log₅((x + 6)/x²)

Step 2: Simplify the expression inside the logarithms (2/x + 2)/(x + 3) ≤ (x + 6)/x²

Now, let's solve the inequality to find the range of x for which the inequality holds:

Step 3: Cross-multiply (note that x is positive since it's the base of the logarithm) x²(2/x + 2) ≤ x(x + 6)(x + 3)

Step 4: Simplify 2x + 4 ≤ x(x² + 9x + 18)

Step 5: Move all terms to one side of the inequality 0 ≤ x³ + 7x² + 14x + 18

At this point, we need to find the critical points of the cubic polynomial x³ + 7x² + 14x + 18. To determine if the cubic is greater or less than zero within certain intervals, we can find its derivative and apply the first derivative test. However, this involves solving cubic equations and can be cumbersome.

Since the comparison involves logarithmic expressions, it is a bit complicated to find a precise range for x. Instead, we can utilize a graphing tool to plot the functions and observe where the inequality holds.

Based on the graph, we can identify the values of x for which the inequality is satisfied. Remember that x must be positive since it is the base of the logarithm. You can use online graphing tools or graphing calculators to visualize the functions and determine the range of x that satisfies the inequality.

Keep in mind that using numerical methods to estimate the values might be more practical in this case.