If `log_(10)2=0.3010...,` the number of digits in the number `2000^(2000)` is

Options

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

Answer

To find the number of digits in the number \( 2000^{2000} \), we can follow these steps:

### Step 1: Express the number
Let \( y = 2000^{2000} \).

### Step 2: Use logarithms to find the number of digits
The number of digits \( d \) in a number \( n \) can be found using the formula:
\[ d = \lfloor \log_{10} n \rfloor + 1 \]
Thus, we need to calculate \( \log_{10} y \).

### Step 3: Apply logarithmic properties
Using the property of logarithms, we can express \( \log_{10} y \) as:
\[ \log_{10} y = \log_{10} (2000^{2000}) = 2000 \cdot \log_{10} (2000) \]

### Step 4: Break down \( \log_{10} (2000) \)
We can express \( 2000 \) as:
\[ 2000 = 2 \times 1000 = 2 \times 10^3 \]
Thus,
\[ \log_{10} (2000) = \log_{10} (2 \times 10^3) = \log_{10} 2 + \log_{10} (10^3) \]
Using the property \( \log_{10} (10^3) = 3 \), we have:
\[ \log_{10} (2000) = \log_{10} 2 + 3 \]

### Step 5: Substitute the known value
We know that \( \log_{10} 2 = 0.3010 \). Therefore:
\[ \log_{10} (2000) = 0.3010 + 3 = 3.3010 \]

### Step 6: Calculate \( \log_{10} y \)
Now substituting back into our equation for \( \log_{10} y \):
\[ \log_{10} y = 2000 \cdot \log_{10} (2000) = 2000 \cdot 3.3010 = 6602 \]

### Step 7: Find the number of digits
Using the formula for the number of digits:
\[ d = \lfloor \log_{10} y \rfloor + 1 = \lfloor 6602 \rfloor + 1 = 6602 + 1 = 6603 \]

### Final Answer
Thus, the number of digits in the number \( 2000^{2000} \) is \( \boxed{6603} \).

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