The value of `log_6 log_5 15625` is

Options

A. 1
B. 2
C. 3
D. none of these

Answer

To solve the problem \( \log_6 \log_5 15625 \), we will break it down step by step.

### Step 1: Calculate \( \log_5 15625 \)

First, we need to express \( 15625 \) as a power of \( 5 \).

\[ 15625 = 5^6 \]

### Step 2: Use the logarithmic property

Now, we can use the property of logarithms that states \( \log_b (a^m) = m \cdot \log_b a \). Here, \( a = 5 \) and \( m = 6 \):

\[ \log_5 15625 = \log_5 (5^6) = 6 \cdot \log_5 5 \]

### Step 3: Simplify \( \log_5 5 \)

Using the property that \( \log_b b = 1 \):

\[ \log_5 5 = 1 \]

### Step 4: Substitute back into the equation

Now substitute back into the equation:

\[ \log_5 15625 = 6 \cdot 1 = 6 \]

### Step 5: Calculate \( \log_6 6 \)

Now we need to find \( \log_6 \log_5 15625 \):

\[ \log_6 \log_5 15625 = \log_6 6 \]

### Step 6: Simplify \( \log_6 6 \)

Again, using the property that \( \log_b b = 1 \):

\[ \log_6 6 = 1 \]

### Final Answer

Thus, the value of \( \log_6 \log_5 15625 \) is:

\[ \boxed{1} \]

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