Let's break down the logarithmic expressions you provided:

1. \( \log_5{25} \)

This is asking, "To what power must 5 be raised to give 25?" We know:
\[
5^2 = 25
\]
Thus:
\[
\log_5{25} = 2
\]

2. \( \log_{25}{625} \)

This is asking, "To what power must 25 be raised to give 625?" Now, remember that:
\[
25 = 5^2 \quad \text{and} \quad 625 = 5^4
\]
Thus, this can be rewritten as:
\[
\log_{25}{625} = \log_{5^2}{5^4}
\]
Using the change of base property:
\[
\log_{a^m}{b^n} = \frac{n}{m}
\]
We get:
\[
\log_{5^2}{5^4} = \frac{4}{2} = 2
\]

So, \( \log_{25}{625} = 2 \).

### Final Results:
1. \( \log_5{25} = 2 \)
2. \( \log_{25}{625} = 2 \)

Would you like any more details or clarifications?

Here are 5 related questions:
1. How do you change the base of a logarithm?
2. What is \( \log_3{27} \)?
3. What is the relationship between exponential and logarithmic functions?
4. How do logarithmic properties simplify calculations?
5. How can we solve logarithmic equations?

**Tip:** Always check if the bases of the logarithms can be expressed as powers of smaller numbers—it can simplify the calculations!

Understand how to solve logarithmic problems like log_5{25} and log_25{625} using logarithmic properties and the change of base theorem. This detailed guide is suitable for students in Grades 10-12 and covers key logarithmic and exponential concepts.

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