How many digits are in the base $10$ representation of $3^{30}$ given $\log 3 = 0.47712$?

[i]2015 CCA Math Bonanza Lightning Round #1.4[/i]
1. To determine the number of digits in \(3^{30}\), we use the formula for the number of digits of a number \(n\) in base 10, which is given by:
   \[
   \text{Number of digits} = \lfloor \log_{10} n \rfloor + 1
   \]
2. We need to find \(\log_{10} (3^{30})\). Using the properties of logarithms, we have:
   \[
   \log_{10} (3^{30}) = 30 \cdot \log_{10} 3
   \]
3. Given \(\log_{10} 3 = 0.47712\), we can substitute this value into the equation:
   \[
   \log_{10} (3^{30}) = 30 \cdot 0.47712 = 14.3136
   \]
4. Now, we apply the formula for the number of digits:
   \[
   \text{Number of digits} = \lfloor 14.3136 \rfloor + 1 = 14 + 1 = 15
   \]

Therefore, \(3^{30}\) has 15 digits.

The final answer is \(\boxed{15}\).