If log 2 = 0.3010, the number of digits in `5^(20)` is:

Options

A. 16
B. 14
C. 8
D. none of these

Answer

To find the number of digits in \( 5^{20} \), we can use the formula for the number of digits \( d \) in a number \( n \), which is given by:

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

In this case, \( n = 5^{20} \). Therefore, we need to calculate \( \log_{10} (5^{20}) \).

### Step 1: Calculate \( \log_{10} (5^{20}) \)

Using the logarithmic identity \( \log_{10} (a^b) = b \cdot \log_{10} a \):

\[ \log_{10} (5^{20}) = 20 \cdot \log_{10} 5 \]

### Step 2: Express \( \log_{10} 5 \) in terms of \( \log_{10} 2 \)

We know that:

\[ \log_{10} 5 = \log_{10} \left( \frac{10}{2} \right) = \log_{10} 10 - \log_{10} 2 \]

Since \( \log_{10} 10 = 1 \) and we are given \( \log_{10} 2 = 0.3010 \):

\[ \log_{10} 5 = 1 - \log_{10} 2 = 1 - 0.3010 = 0.6990 \]

### Step 3: Substitute \( \log_{10} 5 \) back into the equation

Now we substitute \( \log_{10} 5 \) back into our equation for \( \log_{10} (5^{20}) \):

\[ \log_{10} (5^{20}) = 20 \cdot 0.6990 = 13.980 \]

### Step 4: Calculate the number of digits

Now we can calculate the number of digits \( d \):

\[ d = \lfloor 13.980 \rfloor + 1 = 13 + 1 = 14 \]

### Conclusion

Thus, the number of digits in \( 5^{20} \) is **14**.

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