If `log_(10)2=0*30103, log_(10)3=0*47712` , the number of digits in `3^(12)xx2^(8)` is

Options

A. `7`
B. `8`
C. `9`
D. `10`

Answer

To find the number of digits in the number \( N = 3^{12} \times 2^{8} \), we can use the logarithmic properties. The number of digits \( d \) in a number \( N \) can be found using the formula:

\[ d = \lfloor \log_{10} N \rfloor + 1 \]

### Step-by-step Solution:

1. **Express \( N \) in terms of logarithms**:
\[ N = 3^{12} \times 2^{8} \]
Taking the logarithm (base 10) of both sides:
\[ \log_{10} N = \log_{10} (3^{12} \times 2^{8}) \]

2. **Apply the logarithmic property**:
Using the property \( \log_{10} (a \times b) = \log_{10} a + \log_{10} b \) and \( \log_{10} (a^b) = b \cdot \log_{10} a \):
\[ \log_{10} N = \log_{10} (3^{12}) + \log_{10} (2^{8}) = 12 \cdot \log_{10} 3 + 8 \cdot \log_{10} 2 \]

3. **Substitute the given values**:
We know that \( \log_{10} 2 = 0.30103 \) and \( \log_{10} 3 = 0.47712 \):
\[ \log_{10} N = 12 \cdot 0.47712 + 8 \cdot 0.30103 \]

4. **Calculate each term**:
- Calculate \( 12 \cdot 0.47712 \):
\[ 12 \cdot 0.47712 = 5.72544 \]
- Calculate \( 8 \cdot 0.30103 \):
\[ 8 \cdot 0.30103 = 2.40824 \]

5. **Add the results**:
\[ \log_{10} N = 5.72544 + 2.40824 = 8.13368 \]

6. **Find the number of digits**:
Now, using the formula for the number of digits:
\[ d = \lfloor \log_{10} N \rfloor + 1 \]
The integer part \( \lfloor 8.13368 \rfloor = 8 \), so:
\[ d = 8 + 1 = 9 \]

### Conclusion:
The number of digits in \( 3^{12} \times 2^{8} \) is **9**.