Find : (i) the logarithm of 1000 to the base 10.
(ii) the logarithm of `(1)/(9)` to the base 3.

Answer

To solve the given problems, we will break down each part step by step.

### Part (i): Find the logarithm of 1000 to the base 10.

1. **Express 1000 as a power of 10**:
\[ 1000 = 10^3 \]

2. **Apply the logarithm property**:
Using the property of logarithms, \(\log_b(a^c) = c \cdot \log_b(a)\):
\[ \log_{10}(1000) = \log_{10}(10^3) = 3 \cdot \log_{10}(10) \]

3. **Evaluate \(\log_{10}(10)\)**:
Since the base and the number are the same, we know:
\[ \log_{10}(10) = 1 \]

4. **Substitute back into the equation**:
\[ \log_{10}(1000) = 3 \cdot 1 = 3 \]

### Final Answer for Part (i):
\[ \log_{10}(1000) = 3 \]

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### Part (ii): Find the logarithm of \(\frac{1}{9}\) to the base 3.

1. **Express \(\frac{1}{9}\) as a power of 3**:
\[ \frac{1}{9} = \frac{1}{3^2} = 3^{-2} \]

2. **Apply the logarithm property**:
Using the property \(\log_b(a^c) = c \cdot \log_b(a)\):
\[ \log_{3}\left(\frac{1}{9}\right) = \log_{3}(3^{-2}) = -2 \cdot \log_{3}(3) \]

3. **Evaluate \(\log_{3}(3)\)**:
Again, since the base and the number are the same:
\[ \log_{3}(3) = 1 \]

4. **Substitute back into the equation**:
\[ \log_{3}\left(\frac{1}{9}\right) = -2 \cdot 1 = -2 \]

### Final Answer for Part (ii):
\[ \log_{3}\left(\frac{1}{9}\right) = -2 \]

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