Find values of lamda for which `1/log_3lamda+1/log_4lamdagt2` .

Answer

To solve the inequality \( \frac{1}{\log_3 \lambda} + \frac{1}{\log_4 \lambda} > 2 \), we will follow these steps:

### Step 1: Rewrite the logarithms in terms of a common base
We can convert the logarithms to a common base, such as base 10, using the change of base formula:
\[ \log_3 \lambda = \frac{\log_{10} \lambda}{\log_{10} 3} \quad \text{and} \quad \log_4 \lambda = \frac{\log_{10} \lambda}{\log_{10} 4} \]
Thus, we can rewrite the inequality as:
\[ \frac{1}{\frac{\log_{10} \lambda}{\log_{10} 3}} + \frac{1}{\frac{\log_{10} \lambda}{\log_{10} 4}} > 2 \]

### Step 2: Simplify the expression
This simplifies to:
\[ \frac{\log_{10} 3}{\log_{10} \lambda} + \frac{\log_{10} 4}{\log_{10} \lambda} > 2 \]
Combining the fractions gives:
\[ \frac{\log_{10} 3 + \log_{10} 4}{\log_{10} \lambda} > 2 \]

### Step 3: Use properties of logarithms
Using the property of logarithms that states \( \log_a b + \log_a c = \log_a (bc) \), we can combine the logs:
\[ \frac{\log_{10} (3 \cdot 4)}{\log_{10} \lambda} > 2 \]
This simplifies to:
\[ \frac{\log_{10} 12}{\log_{10} \lambda} > 2 \]

### Step 4: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ \log_{10} 12 > 2 \log_{10} \lambda \]

### Step 5: Rewrite the inequality
This can be rewritten as:
\[ \log_{10} 12 > \log_{10} (\lambda^2) \]

### Step 6: Exponentiate both sides
Exponentiating both sides (using base 10) leads to:
\[ 12 > \lambda^2 \]

### Step 7: Solve for \( \lambda \)
Taking the square root of both sides gives:
\[ -\sqrt{12} < \lambda < \sqrt{12} \]
Since \( \lambda \) must be positive (as logarithms are only defined for positive arguments), we restrict our solution to:
\[ 0 < \lambda < \sqrt{12} \]

### Final Answer
Thus, the values of \( \lambda \) for which the inequality holds are:
\[ \lambda \in (0, \sqrt{12}) \]