Number of digits in `60^(12)`

Options

A. 25
B. 22
C. 23
D. 24

Answer

To find the number of digits in \(60^{12}\), we can use the formula for the number of digits \(d\) in a number \(n\), which is given by:

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

### Step-by-Step Solution:

1. **Define the number**: Let \(x = 60^{12}\).

2. **Take the logarithm**: We will take the base 10 logarithm of both sides:
\[ \log_{10} x = \log_{10} (60^{12}) \]

3. **Use the power rule of logarithms**: According to the power rule, we can bring the exponent down:
\[ \log_{10} x = 12 \cdot \log_{10} 60 \]

4. **Break down \(60\)**: We can express \(60\) as \(60 = 2^1 \cdot 3^1 \cdot 10^1\). Therefore, we can use the property of logarithms to break it down further:
\[ \log_{10} 60 = \log_{10} (2 \cdot 3 \cdot 10) = \log_{10} 2 + \log_{10} 3 + \log_{10} 10 \]

5. **Substitute known logarithm values**: We know that:
- \(\log_{10} 2 \approx 0.301\)
- \(\log_{10} 3 \approx 0.477\)
- \(\log_{10} 10 = 1\)

Therefore:
\[ \log_{10} 60 \approx 0.301 + 0.477 + 1 = 1.778 \]

6. **Calculate \(\log_{10} x\)**: Now substitute this back into our equation:
\[ \log_{10} x = 12 \cdot 1.778 \approx 21.336 \]

7. **Find the number of digits**: Now we can find the number of digits using the formula:
\[ d = \lfloor \log_{10} x \rfloor + 1 = \lfloor 21.336 \rfloor + 1 = 21 + 1 = 22 \]

### Final Answer:
The number of digits in \(60^{12}\) is **22**.