To determine the number of digits in \(6^{25}\), we can use the formula for finding the number of digits of a number \(n\), given by:

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

Here, \(n = 6^{25}\).

Using the property of logarithms:

$\log_{10}(a^b) = b \cdot \log_{10} a$

We find:

$\log_{10} (6^{25}) = 25 \cdot \log_{10} 6$

We know that \(6 = 2 \times 3\), thus:

$\log_{10} 6 = \log_{10} (2 \times 3) = \log_{10} 2 + \log_{10} 3$

Substitute the given values for logarithms:

$\log_{10} 6 = 0.3010 + 0.4771 = 0.7781$

Now we can calculate:

$\log_{10} (6^{25}) = 25 \cdot 0.7781 = 19.4525$

The number of digits \(d\) is then:

d = $\lfloor 19.4525 \rfloor$ + 1 = 19 + 1 = 20

Therefore, the number of digits in $\(6^{25}\)$ is 20.