Find the value of the logarithms without using a calculator.

To find the values of the given logarithms without using a calculator, we can use the definition of logarithms and properties of exponents.

a. \( \log_2 8 \)

Since \( 8 = 2^3 \), we can write:

\[
\log_2 8 = \log_2 (2^3) = 3
\]

b. \( \log_5 25 \)

Since \( 25 = 5^2 \), we have:

\[
\log_5 25 = \log_5 (5^2) = 2
\]

c. \( \log_4 64 \)

Since \( 64 = 4^3 \) (because \( 4 = 2^2 \) and \( 64 = (2^2)^3 = 2^6 \)), we can write:

\[
\log_4 64 = \log_4 (4^3) = 3
\]

d. \( \log_3 9 \)

Since \( 9 = 3^2 \), we can write:

\[
\log_3 9 = \log_3 (3^2) = 2
\]

e. \( \log_5 625 \)

Since \( 625 = 5^4 \), we have:

\[
\log_5 625 = \log_5 (5^4) = 4
\]

So, the answers are:

a. \( 3 \)
b. \( 2 \)
c. \( 3 \)
d. \( 2 \)
e. \( 4 \)