If `log_(2)(log_(2)(log_(2)x))=2`, then the number of digits in x, is `(log_(10)2=0.3010)`

Options

A. 7
B. 6
C. 5
D. 4

Explanation

To solve the equation \( \log_{2}(\log_{2}(\log_{2} x)) = 2 \) and find the number of digits in \( x \), we can follow these steps:

### Step 1: Solve the innermost logarithm
Start with the equation:
\[ \log_{2}(\log_{2}(\log_{2} x)) = 2 \]
Using the property of logarithms, we can rewrite this as:
\[ \log_{2}(\log_{2} x) = 2^2 = 4 \]

### Step 2: Solve the next logarithm
Now, we apply the logarithmic property again:
\[ \log_{2} x = 2^4 = 16 \]

### Step 3: Solve for \( x \)
Next, we apply the logarithmic property one more time:
\[ x = 2^{16} \]

### Step 4: Find the number of digits in \( x \)
To find the number of digits in \( x \), we can use the formula for the number of digits \( d \) in a number \( n \):
\[ d = \lfloor \log_{10} n \rfloor + 1 \]
Substituting \( n = x = 2^{16} \):
\[ d = \lfloor \log_{10}(2^{16}) \rfloor + 1 \]
Using the property of logarithms:
\[ \log_{10}(2^{16}) = 16 \cdot \log_{10}(2) \]
Given that \( \log_{10}(2) = 0.3010 \):
\[ \log_{10}(2^{16}) = 16 \cdot 0.3010 = 4.816 \]
Now, applying the floor function:
\[ d = \lfloor 4.816 \rfloor + 1 = 4 + 1 = 5 \]

### Conclusion
Thus, the number of digits in \( x \) is \( 5 \).