In which of the following intervals does `(1)/(log_(1//2) (1/3))+(1)/(log_(1//5)(1//3))` lies
i) (1,2)
ii) (2,3)
iii) (3,4)
iv) (4,5)

Options

A. (1, 2)
B. (2, 3 )
C. (3, 4)
D. (4, 5)

Answer

To solve the expression \(\frac{1}{\log_{1/2}(1/3)} + \frac{1}{\log_{1/5}(1/3)}\), we will simplify it step by step.

### Step 1: Rewrite the logarithms using the change of base formula
Using the change of base formula, we can rewrite the logarithms:

\[ \log_{1/2}(1/3) = \frac{\log_e(1/3)}{\log_e(1/2)} \]
\[ \log_{1/5}(1/3) = \frac{\log_e(1/3)}{\log_e(1/5)} \]

### Step 2: Substitute the rewritten logarithms into the expression
Now, we substitute these into the original expression:

\[ \frac{1}{\log_{1/2}(1/3)} = \frac{\log_e(1/2)}{\log_e(1/3)} \]
\[ \frac{1}{\log_{1/5}(1/3)} = \frac{\log_e(1/5)}{\log_e(1/3)} \]

So, the expression becomes:

\[ \frac{\log_e(1/2)}{\log_e(1/3)} + \frac{\log_e(1/5)}{\log_e(1/3)} \]

### Step 3: Combine the fractions
Now, we can combine these fractions:

\[ \frac{\log_e(1/2) + \log_e(1/5)}{\log_e(1/3)} \]

### Step 4: Use the property of logarithms
Using the property of logarithms that states \(\log_a(b) + \log_a(c) = \log_a(bc)\):

\[ \log_e(1/2) + \log_e(1/5) = \log_e\left(\frac{1}{2} \cdot \frac{1}{5}\right) = \log_e\left(\frac{1}{10}\right) \]

Thus, we can rewrite our expression as:

\[ \frac{\log_e(1/10)}{\log_e(1/3)} \]

### Step 5: Rewrite the logarithm
We can express \(\log_e(1/10)\) as \(-\log_e(10)\):

\[ \frac{-\log_e(10)}{\log_e(1/3)} = -\frac{\log_e(10)}{\log_e(1/3)} \]

### Step 6: Calculate the value
Now, we need to find the numerical value of \(-\frac{\log_e(10)}{\log_e(1/3)}\).

Using approximate values:
- \(\log_e(10) \approx 2.302\)
- \(\log_e(1/3) \approx -1.099\)

Thus:

\[ -\frac{2.302}{-1.099} \approx 2.1 \]

### Conclusion
The value of the expression \(\frac{1}{\log_{1/2}(1/3)} + \frac{1}{\log_{1/5}(1/3)}\) is approximately \(2.1\), which lies in the interval (2, 3).

### Final Answer
The correct option is **ii) (2, 3)**.

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