Solve `((1)/(2))^(logx^(2))+2gt3.2^(log(-x))`

Answer

To solve the inequality \(\left(\frac{1}{2}\right)^{\log x^2} + 2 > 3 \cdot 2^{\log(-x)}\), we will proceed step by step.

### Step 1: Rewrite the terms using properties of logarithms
We know that \(\log x^2 = 2 \log x\). Therefore, we can rewrite the left-hand side:
\[ \left(\frac{1}{2}\right)^{\log x^2} = \left(\frac{1}{2}\right)^{2 \log x} = \left(\left(\frac{1}{2}\right)^{\log x}\right)^2 \]
Thus, the inequality becomes:
\[ \left(\left(\frac{1}{2}\right)^{\log x}\right)^2 + 2 > 3 \cdot 2^{\log(-x)} \]

### Step 2: Simplify the right-hand side
Using the property \(a^{\log b} = b^{\log a}\), we can rewrite \(2^{\log(-x)}\) as:
\[ 2^{\log(-x)} = -x^{\log 2} \]
So the inequality now looks like:
\[ \left(\left(\frac{1}{2}\right)^{\log x}\right)^2 + 2 > 3(-x^{\log 2}) \]

### Step 3: Let \(t = \left(\frac{1}{2}\right)^{\log(-x)}\)
Let \(t = \left(\frac{1}{2}\right)^{\log(-x)}\). Then we can express the inequality as:
\[ t^2 + 2 > 3(-x^{\log 2}) \]

### Step 4: Rearranging the inequality
Rearranging gives us:
\[ t^2 - 3(-x^{\log 2}) + 2 > 0 \]

### Step 5: Solve the quadratic inequality
The quadratic \(t^2 - 3t + 2 = 0\) can be factored as:
\[ (t - 1)(t - 2) > 0 \]
The roots are \(t = 1\) and \(t = 2\).

### Step 6: Analyze the intervals
The sign of the quadratic changes at the roots. Thus, we analyze the intervals:
- For \(t < 1\), the expression is positive.
- For \(1 < t < 2\), the expression is negative.
- For \(t > 2\), the expression is positive.

So, the solution for \(t\) is:
\[ t < 1 \quad \text{or} \quad t > 2 \]

### Step 7: Substitute back for \(t\)
Substituting back for \(t\):
1. For \(t < 1\):
\[ \left(\frac{1}{2}\right)^{\log(-x)} < 1 \implies \log(-x) < 0 \implies -x < 1 \implies x > -1 \]
2. For \(t > 2\):
\[ \left(\frac{1}{2}\right)^{\log(-x)} > 2 \implies \log(-x) < -1 \implies -x < \frac{1}{2} \implies x > -\frac{1}{2} \]

### Final Solution
Combining these results, we have:
\[ x > -1 \quad \text{and} \quad x > -\frac{1}{2} \]
Thus, the solution set is:
\[ x > -\frac{1}{2} \]