How many positive integers b have the property that `log_(b)729` is a positive integer ?

Answer

To solve the problem of how many positive integers \( b \) have the property that \( \log_b 729 \) is a positive integer, we can follow these steps:

### Step 1: Understand the condition
We need to find \( b \) such that \( \log_b 729 \) is a positive integer. This means that \( 729 \) must be expressible as \( b^k \) for some positive integer \( k \).

### Step 2: Express 729 in terms of its prime factors
First, we express \( 729 \) in terms of its prime factors. We know that:
\[ 729 = 3^6 \]

### Step 3: Set up the equation
Since \( 729 = b^k \), we can write:
\[ b^k = 3^6 \]
This implies that \( b \) must also be a power of \( 3 \). Let's express \( b \) as:
\[ b = 3^m \]
for some positive integer \( m \).

### Step 4: Substitute \( b \) into the equation
Substituting \( b \) into the equation gives us:
\[ (3^m)^k = 3^6 \]
This simplifies to:
\[ 3^{mk} = 3^6 \]

### Step 5: Equate the exponents
Since the bases are the same, we can equate the exponents:
\[ mk = 6 \]

### Step 6: Find the positive integer pairs \((m, k)\)
Now we need to find the positive integer pairs \((m, k)\) such that \( mk = 6 \). The pairs of positive integers that satisfy this equation are:
1. \( (1, 6) \)
2. \( (2, 3) \)
3. \( (3, 2) \)
4. \( (6, 1) \)

### Step 7: Determine the corresponding values of \( b \)
Now we can find the corresponding values of \( b \):
1. For \( (1, 6) \): \( b = 3^1 = 3 \)
2. For \( (2, 3) \): \( b = 3^2 = 9 \)
3. For \( (3, 2) \): \( b = 3^3 = 27 \)
4. For \( (6, 1) \): \( b = 3^6 = 729 \)

### Step 8: Count the valid values of \( b \)
The valid values of \( b \) that we found are \( 3, 9, 27, \) and \( 729 \). Therefore, there are 4 positive integers \( b \) such that \( \log_b 729 \) is a positive integer.

### Final Answer
The number of positive integers \( b \) such that \( \log_b 729 \) is a positive integer is:
\[ \boxed{4} \]