Question

1)log^3(7-x)>81 2)log^1/3(x^2-x)>1

Answer 1

Вы наверное подразумевает, что основание логарифм 3?
Тогда так:
1) {7-х >4; 7-х >0
{ х < 3; х <7
Ответ: х <3

Answer 2

To solve these logarithmic inequalities, we'll break them down step by step. Remember that when dealing with logarithmic inequalities, we have to be cautious of the domain of the logarithmic expressions.

1. log^3(7 - x) > 81:

Step 1: Eliminate the logarithmic base 3 by using exponentiation. We raise 3 to the power of both sides: 3^(log^3(7 - x)) > 3^81

Step 2: Simplify the left side using the property: a^(log_a(b)) = b 7 - x > 3^81

Step 3: Isolate x on one side of the inequality: x < 7 - 3^81

So the solution to the inequality is x < 7 - 3^81.

2. log^(1/3)(x^2 - x) > 1:

Step 1: Eliminate the logarithmic base 1/3 by using exponentiation. We raise 1/3 to the power of both sides: (1/3)^(log^(1/3)(x^2 - x)) > (1/3)^1

Step 2: Simplify the left side using the property: a^(log_a(b)) = b x^2 - x > 1/3

Step 3: Move all terms to one side of the inequality: x^2 - x - 1/3 > 0

Step 4: To solve this quadratic inequality, we can find the roots of the quadratic equation x^2 - x - 1/3 = 0: x = [1 ± √(1 - 4(-1/3))] / 2 x = [1 ± √(1 + 4/3)] / 2 x = [1 ± √(7/3)] / 2

Step 5: Analyze the inequality by considering the sign of the expression (x^2 - x - 1/3) in different intervals:

• Interval 1: x < (1 - √(7/3)) / 2
• Interval 2: (1 - √(7/3)) / 2 < x < (1 + √(7/3)) / 2
• Interval 3: x > (1 + √(7/3)) / 2

Step 6: Test points within each interval to determine the sign of (x^2 - x - 1/3):

• For x < (1 - √(7/3)) / 2, choose x = 0 (a value less than the first interval's lower bound). (0)^2 - 0 - 1/3 = -1/3 (negative)

• For (1 - √(7/3)) / 2 < x < (1 + √(7/3)) / 2, choose x = 1 (a value within the interval). (1)^2 - 1 - 1/3 = -1/3 (negative)

• For x > (1 + √(7/3)) / 2, choose x = 2 (a value greater than the last interval's upper bound). (2)^2 - 2 - 1/3 = 7/3 (positive)

Step 7: Based on the analysis, the inequality x^2 - x - 1/3 > 0 is satisfied for x in the interval x > (1 + √(7/3)) / 2.

So the solution to the inequality is x > (1 + √(7/3)) / 2.