What is the value of digits in `3^(3^(3))`? Given that log 3 = 0.47712?

Answer

To find the number of digits in \( 3^{3^{3}} \), we can follow these steps:

### Step 1: Simplify the expression
Let \( x = 3^{3^{3}} \). We can rewrite \( 3^{3^{3}} \) as \( 3^{27} \) since \( 3^{3} = 27 \).

### 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:
\[ d = \lfloor \log_{10} n \rfloor + 1 \]
In our case, we need to find \( d = \lfloor \log_{10} (3^{27}) \rfloor + 1 \).

### Step 3: Apply logarithm properties
Using the property of logarithms, we can simplify:
\[ \log_{10} (3^{27}) = 27 \cdot \log_{10} (3) \]
Given that \( \log_{10} (3) = 0.47712 \), we can substitute this value in:
\[ \log_{10} (3^{27}) = 27 \cdot 0.47712 \]

### Step 4: Calculate the logarithm
Now, we calculate:
\[ 27 \cdot 0.47712 = 12.86724 \]

### Step 5: Find the number of digits
Now, we apply the formula for the number of digits:
\[ d = \lfloor 12.86724 \rfloor + 1 = 12 + 1 = 13 \]

### Final Answer
Thus, the number of digits in \( 3^{3^{3}} \) is **13**.

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