The interval of values of x for which 0 < log(x^2 - 2x - 2) < 1 is:
x ∈ (-1, 1 - √3) ∪ (1 + √3, 3)
The interval of values of x for which 0 < log(x^2 - 2x - 2) < 1:
Step 1: Rewrite the inequalities.
Since the logarithm is only defined for positive values, we know that x^2 - 2x - 2 > 0. Additionally, the inequality 0 < log(x^2 - 2x - 2) < 1 implies that 1 < x^2 - 2x - 2 < 10.
Step 2: Solve the quadratic equation.
The inequality x^2 - 2x - 2 > 0 can be solved using the quadratic formula
x = (2 ± √(2^2 + 4 * 2 * 2)) / 2 = (2 ± √12) / 2 = 1 ± √3
Therefore, x^2 - 2x - 2 > 0 for x < 1 - √3 or x > 1 + √3.
Step 3: Solve the other inequality.
The inequality 1 < x^2 - 2x - 2 < 10 can be rewritten as:
0 < x^2 - 2x - 3 < 9
(x - 3)(x + 1) < 0
This inequality is true when x is between -1 and 3, exclusive.
Step 4: Combine the solutions.
Combining the solutions from steps 2 and 3, we have:
x ∈ (-∞, 1 - √3) ∪ (1 + √3, ∞) ∩ (-1, 3)
Therefore, the interval of values of x for which 0 < log(x^2 - 2x - 2) < 1 is:
x ∈ (-1, 1 - √3) ∪ (1 + √3, 3)