If `log2=0.30103`, find the number of digits in `2^(56)`

Options

A. A)17
B. B)31
C. C)100
D. D)200

Answer

To find the number of digits in \( 2^{56} \), we can use the logarithmic property that relates the number of digits of a number \( n \) to its logarithm. The formula to find the number of digits \( d \) in a number \( n \) is given by:

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

Here are the steps to solve the problem:

### Step 1: Define the number
Let \( y = 2^{56} \).

### Step 2: Take the logarithm
We need to find \( \log_{10} y \):
\[ \log_{10} y = \log_{10} (2^{56}) \]

### Step 3: Use the power rule of logarithms
Using the property of logarithms, we can rewrite this as:
\[ \log_{10} (2^{56}) = 56 \cdot \log_{10} 2 \]

### Step 4: Substitute the value of \( \log_{10} 2 \)
We know from the problem that \( \log_{10} 2 = 0.30103 \). Therefore:
\[ \log_{10} (2^{56}) = 56 \cdot 0.30103 \]

### Step 5: Calculate the logarithm
Now, we perform the multiplication:
\[ 56 \cdot 0.30103 = 16.85668 \]

### Step 6: Find the number of digits
Now, we apply the formula for the number of digits:
\[ d = \lfloor 16.85668 \rfloor + 1 \]
Calculating the floor function:
\[ \lfloor 16.85668 \rfloor = 16 \]
Thus, the number of digits is:
\[ d = 16 + 1 = 17 \]

### Final Answer
The number of digits in \( 2^{56} \) is **17**.

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