To evaluate $\log_{25} 125$, we need to express 125 and/or 25 in terms of common bases, preferably powers of the same number.

Step-by-Step Solution

1. Rewrite the bases:

   • $25 = 5^2$
   • $125 = 5^3$

2. Rewrite the logarithmic expression: $\log_{25} 125 = \log_{5^2} 5^3$

3. Use the change of base formula: When dealing with a logarithm in the form $\log_{a^m} b^n$, we can use the property: $\log_{a^m} b^n = \frac{n}{m} \log_a b$

   Applying this here: $\log_{5^2} 5^3 = \frac{3}{2} \log_5 5$

4. Evaluate $\log_5 5$: Since $\log_5 5 = 1$, we get: $\frac{3}{2} \cdot 1 = \frac{3}{2}$

Final Answer

$\log_{25} 125 = \frac{3}{2}$

Would you like a detailed breakdown of any step?

Here are some related questions to further explore this concept:

1. How would you evaluate $\log_{125} 25$?
2. What is the value of $\log_5 125$?
3. How do properties of logarithms simplify expressions like $\log_{a^n} b^m$?
4. Can you evaluate $\log_{4} 16$ using a similar method?
5. How would you approach $\log_{9} 27$ using common bases?

Tip: Familiarize yourself with properties like $\log_{a^m} b^n = \frac{n}{m} \log_a b$; they help simplify complex logarithmic expressions quickly!