Given that `log(2)=0. 3010 ,` the number of digits in the number `2000^(2000)` is 6601 (b) 6602 (c) 6603 (d) 6604

Options

A. 6601
B. 6602
C. 6603
D. 6604

Explanation

To find the number of digits in the number \( 2000^{2000} \), we can use the formula for the number of digits \( d \) of a number \( n \), which is given by:

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

### Step-by-step Solution:

1. **Identify the number**:
We have \( n = 2000^{2000} \).

2. **Take the logarithm**:
We need to find \( \log_{10}(2000^{2000}) \).
Using the property of logarithms \( \log(a^b) = b \cdot \log(a) \):
\[ \log_{10}(2000^{2000}) = 2000 \cdot \log_{10}(2000) \]

3. **Break down \( 2000 \)**:
We can express \( 2000 \) as \( 2000 = 2 \times 10^3 \).
Therefore, we can write:
\[ \log_{10}(2000) = \log_{10}(2 \times 10^3) = \log_{10}(2) + \log_{10}(10^3) \]
Using the property \( \log(a \cdot b) = \log(a) + \log(b) \) and knowing that \( \log_{10}(10^3) = 3 \):
\[ \log_{10}(2000) = \log_{10}(2) + 3 \]

4. **Substituting the known value**:
We are given \( \log_{10}(2) = 0.3010 \):
\[ \log_{10}(2000) = 0.3010 + 3 = 3.3010 \]

5. **Calculate \( \log_{10}(2000^{2000}) \)**:
Now substituting back:
\[ \log_{10}(2000^{2000}) = 2000 \cdot 3.3010 = 6602 \]

6. **Find the number of digits**:
Using the formula for the number of digits:
\[ d = \lfloor 6602 \rfloor + 1 = 6602 + 1 = 6603 \]

### Final Answer:
The number of digits in \( 2000^{2000} \) is \( 6603 \).