Solve the following inequation: `log_(1//3)xltlog_(1//2)x`

Answer

To solve the inequation \( \log_{\frac{1}{3}} x < \log_{\frac{1}{2}} x \), we will follow these steps:

### Step 1: Rewrite the Inequation
We start with the given inequation:
\[ \log_{\frac{1}{3}} x < \log_{\frac{1}{2}} x \]

### Step 2: Use the Change of Base Formula
Using the change of base formula, we can express the logarithms in terms of natural logarithms:
\[ \log_{\frac{1}{3}} x = \frac{\log_e x}{\log_e \frac{1}{3}} \quad \text{and} \quad \log_{\frac{1}{2}} x = \frac{\log_e x}{\log_e \frac{1}{2}} \]
Substituting these into the inequation gives:
\[ \frac{\log_e x}{\log_e \frac{1}{3}} < \frac{\log_e x}{\log_e \frac{1}{2}} \]

### Step 3: Rearranging the Inequation
To eliminate the fractions, we can multiply both sides by \( \log_e \frac{1}{3} \cdot \log_e \frac{1}{2} \). Since \( \log_e \frac{1}{3} < 0 \) and \( \log_e \frac{1}{2} < 0 \), the direction of the inequality will reverse:
\[ \log_e x \cdot \log_e \frac{1}{2} > \log_e x \cdot \log_e \frac{1}{3} \]

### Step 4: Factor Out \( \log_e x \)
Rearranging gives us:
\[ \log_e x \cdot \left( \log_e \frac{1}{2} - \log_e \frac{1}{3} \right) > 0 \]
Now, we can simplify \( \log_e \frac{1}{2} - \log_e \frac{1}{3} \):
\[ \log_e \frac{1}{2} - \log_e \frac{1}{3} = \log_e \frac{\frac{1}{2}}{\frac{1}{3}} = \log_e \frac{3}{2} \]
Thus, we have:
\[ \log_e x \cdot \log_e \frac{3}{2} > 0 \]

### Step 5: Analyze the Inequality
Since \( \log_e \frac{3}{2} > 0 \) (because \( \frac{3}{2} > 1 \)), we can divide both sides by \( \log_e \frac{3}{2} \) without changing the inequality:
\[ \log_e x > 0 \]

### Step 6: Solve for \( x \)
The inequality \( \log_e x > 0 \) implies:
\[ x > 1 \]

### Final Solution
Thus, the solution to the inequation \( \log_{\frac{1}{3}} x < \log_{\frac{1}{2}} x \) is:
\[ x > 1 \]