Question

Log1/3 (x2-6x+18)-2log1/3(x-4)<0

Answer 1

Ответ:

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Объяснение:

Answer 2

To solve the inequality, we'll apply logarithmic properties to simplify the expression first. Then we'll solve the resulting equation. Let's start by simplifying the expression step by step:

1. Applying the power rule of logarithms: log1/3((x^2 - 6x + 18)) - 2log1/3(x - 4) < 0

2. Using the rule log_b(x) - log_b(y) = log_b(x/y): log1/3((x^2 - 6x + 18)/(x - 4)^2) < 0

3. Rewriting the expression without logarithms: (x^2 - 6x + 18)/(x - 4)^2 < 1/3

Now that we have simplified the expression, we'll solve the equation:

1. Multiplying both sides of the inequality by (x - 4)^2 to eliminate the denominator: (x^2 - 6x + 18) < (x - 4)^2/3

2. Multiplying both sides by 3 to eliminate the fraction: 3(x^2 - 6x + 18) < (x - 4)^2

3. Expanding (x - 4)^2: 3(x^2 - 6x + 18) < x^2 - 8x + 16

4. Distributing 3 to the terms inside the parentheses: 3x^2 - 18x + 54 < x^2 - 8x + 16

5. Rearranging the equation by bringing all terms to one side: 2x^2 - 10x + 38 < 0

Now we have a quadratic inequality. To solve it, we can use various methods such as factoring, completing the square, or using the quadratic formula. However, upon inspecting the discriminant (b^2 - 4ac) of the quadratic equation (2x^2 - 10x + 38), we find that it is negative. This indicates that the quadratic has no real solutions.

Therefore, the inequality log1/3((x^2 - 6x + 18)) - 2log1/3(x - 4) < 0 has no solutions in the real number domain.