If log 27 = 1.431, find the value of :
(i) log 9
(ii) log 300

Answer

To solve the problem, we need to find the values of \( \log 9 \) and \( \log 300 \) given that \( \log 27 = 1.431 \).

### Step-by-Step Solution:

**(i) Finding \( \log 9 \)**

1. **Express 27 in terms of 3**:
We know that \( 27 = 3^3 \).
Therefore, we can write:
\[ \log 27 = \log(3^3) \]

2. **Apply the logarithmic property**:
Using the property \( \log(a^b) = b \cdot \log a \), we have:
\[ \log(3^3) = 3 \cdot \log 3 \]
Thus, we can equate:
\[ 3 \cdot \log 3 = 1.431 \]

3. **Solve for \( \log 3 \)**:
Dividing both sides by 3 gives:
\[ \log 3 = \frac{1.431}{3} = 0.477 \]

4. **Express 9 in terms of 3**:
We know that \( 9 = 3^2 \).
Therefore, we can write:
\[ \log 9 = \log(3^2) \]

5. **Apply the logarithmic property again**:
Using the same property:
\[ \log(3^2) = 2 \cdot \log 3 \]

6. **Substitute the value of \( \log 3 \)**:
Now substituting \( \log 3 = 0.477 \):
\[ \log 9 = 2 \cdot 0.477 = 0.954 \]

**Final answer for (i)**:
\[ \log 9 = 0.954 \]

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**(ii) Finding \( \log 300 \)**

1. **Express 300 in terms of its factors**:
We can express \( 300 \) as \( 300 = 3 \times 100 \).
Thus, we can write:
\[ \log 300 = \log(3 \times 100) \]

2. **Apply the logarithmic property for products**:
Using the property \( \log(m \times n) = \log m + \log n \):
\[ \log 300 = \log 3 + \log 100 \]

3. **Substitute the value of \( \log 3 \)**:
We already found \( \log 3 = 0.477 \), so:
\[ \log 300 = 0.477 + \log 100 \]

4. **Express 100 in terms of powers of 10**:
We know that \( 100 = 10^2 \), so:
\[ \log 100 = \log(10^2) \]

5. **Apply the logarithmic property again**:
Using the property \( \log(a^b) = b \cdot \log a \):
\[ \log(10^2) = 2 \cdot \log 10 \]

6. **Use the known value of \( \log 10 \)**:
We know that \( \log 10 = 1 \):
\[ \log 100 = 2 \cdot 1 = 2 \]

7. **Combine the results**:
Now substituting back:
\[ \log 300 = 0.477 + 2 = 2.477 \]

**Final answer for (ii)**:
\[ \log 300 = 2.477 \]

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### Summary of Answers:
- \( \log 9 = 0.954 \)
- \( \log 300 = 2.477 \)