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!