Question

Log√2 (6-x ^(2)) > 2

Answer 1

ОДЗ : 6 - x² > 0
x² - 6 < 0
(x - √6)(x + √6) < 0
x ∈ (- √6 ; √6)

$log _{ \sqrt{2} }(6- x^{2} )\ \textgreater \ 2\\\\6- x^{2} \ \textgreater \ ( \sqrt{2}) ^{2} \\\\ 6- x^{2} \ \textgreater \ 2\\\\4- x^{2}\ \textgreater \ 0\\\\ x^{2} -4\ \textless \ 0$
$(x-2)(x+2)\ \textless \ 0$
          +                      -                       +
___________₀___________₀____________
                     -2                      2
x ∈ (- 2 ; 2)

Answer 2

To solve the inequality, follow these steps:

Step 1: Move the constant term to the other side of the inequality: Log√2 (6 - x^2) - 2 > 0

Step 2: Combine the logarithmic terms using properties of logarithms. The rule we'll use is: Log_b(A) - Log_b(C) = Log_b(A/C) Log√2 ((6 - x^2) / √2^2) > 0

Step 3: Simplify inside the logarithm: Log√2 ((6 - x^2) / 2) > 0

Step 4: Remove the logarithm by exponentiating both sides. For any positive base "a", if Log_a(B) > 0, then B > 1. √2 ((6 - x^2) / 2) > 1

Step 5: Multiply both sides by 2/√2 to eliminate the square root: (2/√2) * √2 ((6 - x^2) / 2) > (2/√2) * 1

Step 6: Simplify the left side: 2(6 - x^2) > √2

Step 7: Distribute the 2 on the left side: 12 - 2x^2 > √2

Step 8: Move √2 to the other side of the inequality: 12 - √2 > 2x^2

Step 9: Divide both sides by 2 to solve for x^2: (12 - √2) / 2 > x^2

Step 10: Simplify the left side: 6 - (√2 / 2) > x^2

Step 11: Finally, take the square root of both sides. Since the square root is always positive, there is no need to consider the negative square root: √(6 - (√2 / 2)) > x

The solution to the inequality is: x < √(6 - (√2 / 2))

Approximately: x < 2.193