Finding the Number of Digits in $3^{16}$

To find the number of digits in a positive integer $N$ when expressed in decimal form, we use the formula: Number of digits = $ \lfloor \log_{10}(N) \rfloor + 1 $.

Applying the Logarithm Formula

In this problem, the number is $ N = 3^{16} $.

We need to calculate $ \log_{10}(3^{16}) $. Using the logarithm power rule $ \log(a^b) = b \times \log(a) $, we get:

$ \log_{10}(3^{16}) = 16 \times \log_{10}(3) $

Calculating the Logarithmic Value

We use the approximate value of $ \log_{10}(3) \approx 0.4771 $.

$ 16 \times \log_{10}(3) \approx 16 \times 0.4771 $

$ 16 \times 0.4771 = 7.6336 $

Determining the Number of Digits

Now, we apply the formula for the number of digits:

Number of digits = $ \lfloor \log_{10}(3^{16}) \rfloor + 1 $

Number of digits = $ \lfloor 7.6336 \rfloor + 1 $

The floor of $ 7.6336 $ is $ 7 $.

Number of digits = $ 7 + 1 = 8 $

Therefore, $ 3^{16} $ has 8 digits in its decimal form.