The value of `log_(2)log_(3) log_(5)(125)^(3)` is:

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the expression \( \log_{2}(\log_{3}(\log_{5}(125)^{3})) \), we will break it down step by step.

### Step 1: Simplify \( \log_{5}(125)^{3} \)

First, we need to simplify \( \log_{5}(125)^{3} \).

We know that \( 125 = 5^{3} \). Therefore, we can rewrite \( 125^{3} \) as:

\[ 125^{3} = (5^{3})^{3} = 5^{9} \]

Now, we can find \( \log_{5}(5^{9}) \):

\[ \log_{5}(5^{9}) = 9 \]

### Step 2: Substitute back into the expression

Now we substitute this result back into the expression:

\[ \log_{3}(\log_{5}(125)^{3}) = \log_{3}(9) \]

### Step 3: Simplify \( \log_{3}(9) \)

Next, we simplify \( \log_{3}(9) \). We know that \( 9 = 3^{2} \), so:

\[ \log_{3}(9) = \log_{3}(3^{2}) = 2 \]

### Step 4: Substitute back into the expression

Now we substitute this result back into the expression:

\[ \log_{2}(\log_{3}(\log_{5}(125)^{3})) = \log_{2}(2) \]

### Step 5: Simplify \( \log_{2}(2) \)

Finally, we simplify \( \log_{2}(2) \):

\[ \log_{2}(2) = 1 \]

### Final Answer

Thus, the value of \( \log_{2}(\log_{3}(\log_{5}(125)^{3})) \) is:

\[ \boxed{1} \]