Find the value of ` log_(5) log_(2)log_(3) log_(2) 512`.

Explanation

To find the value of \( \log_{5} \log_{2} \log_{3} \log_{2} 512 \), we will follow these steps:

### Step 1: Factorize 512
First, we need to express 512 in terms of powers of 2.
\[ 512 = 2^9 \]

### Step 2: Apply the logarithm property
Using the property of logarithms that states \( \log_{b}(m^n) = n \cdot \log_{b}(m) \), we can rewrite \( \log_{2} 512 \):
\[ \log_{2} 512 = \log_{2} (2^9) = 9 \cdot \log_{2} 2 \]

### Step 3: Simplify \( \log_{2} 2 \)
Since \( \log_{2} 2 = 1 \):
\[ \log_{2} 512 = 9 \cdot 1 = 9 \]

### Step 4: Substitute back into the expression
Now we substitute this value back into the expression:
\[ \log_{5} \log_{2} \log_{3} 9 \]

### Step 5: Simplify \( \log_{3} 9 \)
Next, we need to evaluate \( \log_{3} 9 \):
\[ 9 = 3^2 \implies \log_{3} 9 = \log_{3} (3^2) = 2 \cdot \log_{3} 3 \]
Since \( \log_{3} 3 = 1 \):
\[ \log_{3} 9 = 2 \cdot 1 = 2 \]

### Step 6: Substitute again
Now we substitute this value back:
\[ \log_{5} \log_{2} 2 \]

### Step 7: Simplify \( \log_{2} 2 \)
Again, since \( \log_{2} 2 = 1 \):
\[ \log_{5} 1 \]

### Step 8: Evaluate \( \log_{5} 1 \)
Finally, we know that:
\[ \log_{b} 1 = 0 \quad \text{for any base } b \neq 1 \]
Thus:
\[ \log_{5} 1 = 0 \]

### Final Answer
The value of \( \log_{5} \log_{2} \log_{3} \log_{2} 512 \) is:
\[ \boxed{0} \]