if log 3 = 0.4711, find the number of digits in `3^(43)`

Answer

To find the number of digits in \(3^{43}\), we can use the formula that relates the number of digits \(d\) of a number \(n\) to its logarithm:

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

In this case, we want to find \(d\) for \(n = 3^{43}\).

### Step-by-Step Solution:

1. **Use the logarithmic identity**:
\[ \log_{10} (3^{43}) = 43 \cdot \log_{10} 3 \]

2. **Substitute the given value**:
We know from the problem that \(\log_{10} 3 = 0.4711\). Therefore,
\[ \log_{10} (3^{43}) = 43 \cdot 0.4711 \]

3. **Calculate the product**:
\[ 43 \cdot 0.4711 = 20.2573 \]

4. **Find the number of digits**:
Now, we can use the formula for the number of digits:
\[ d = \lfloor 20.2573 \rfloor + 1 \]
The floor function \(\lfloor 20.2573 \rfloor\) gives us 20. Therefore,
\[ d = 20 + 1 = 21 \]

Thus, the number of digits in \(3^{43}\) is **21**.