Find the value of `log_(s) log_(s) (3125)`

Answer

To find the value of \( \log_{5} \log_{5} (3125) \), we will follow these steps:

### Step 1: Simplify \( 3125 \)
First, we need to express \( 3125 \) in terms of powers of \( 5 \).

We can factor \( 3125 \) as follows:
\[ 3125 = 5 \times 625 \]
Continuing to factor \( 625 \):
\[ 625 = 5 \times 125 \]
And \( 125 \):
\[ 125 = 5 \times 25 \]
And \( 25 \):
\[ 25 = 5 \times 5 \]
Thus, we can express \( 3125 \) as:
\[ 3125 = 5 \times 5 \times 5 \times 5 \times 5 = 5^5 \]

### Step 2: Calculate \( \log_{5} (3125) \)
Now, we can calculate \( \log_{5} (3125) \):
\[ \log_{5} (3125) = \log_{5} (5^5) \]
Using the logarithmic property \( \log_{a} (a^b) = b \):
\[ \log_{5} (5^5) = 5 \]

### Step 3: Calculate \( \log_{5} \log_{5} (3125) \)
Now we substitute back into the original expression:
\[ \log_{5} \log_{5} (3125) = \log_{5} (5) \]
Using the property \( \log_{a} (a) = 1 \):
\[ \log_{5} (5) = 1 \]

### Final Answer
Thus, the value of \( \log_{5} \log_{5} (3125) \) is:
\[ \boxed{1} \]

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