Prove that: `2//5ltlog_(10)3lt1//2`

Answer

To prove that \( \frac{2}{5} < \log_{10} 3 < \frac{1}{2} \), we will follow these steps:

### Step 1: Calculate the value of \( \log_{10} 3 \)

Using a scientific calculator or logarithm table, we find:

\[ \log_{10} 3 \approx 0.4771 \]

### Step 2: Convert \( \frac{2}{5} \) and \( \frac{1}{2} \) to decimal form

Now, we will convert \( \frac{2}{5} \) and \( \frac{1}{2} \) into decimal values.

\[ \frac{2}{5} = 0.4 \]
\[ \frac{1}{2} = 0.5 \]

### Step 3: Compare the values

Now we compare the values:

1. Check if \( \frac{2}{5} < \log_{10} 3 \):
\[ 0.4 < 0.4771 \quad \text{(True)} \]

2. Check if \( \log_{10} 3 < \frac{1}{2} \):
\[ 0.4771 < 0.5 \quad \text{(True)} \]

### Conclusion

Since both inequalities are true, we conclude that:

\[ \frac{2}{5} < \log_{10} 3 < \frac{1}{2} \]

Thus, we have proved the statement.

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