If two distinct numbers a and be are selected from the set `{5^(1), 5^(2), 5^(3)……….5^(9)}`, then the probability that `log_(a)b` is an integer is

Options

A. `(7)/(18)`
B. `(5)/(9)`
C. `(7)/(36)`
D. `(3)/(16)`

Answer

To solve the problem, we need to find the probability that \( \log_a b \) is an integer when two distinct numbers \( a \) and \( b \) are selected from the set \( \{ 5^1, 5^2, 5^3, \ldots, 5^9 \} \).

### Step-by-step Solution:

1. **Identify the Set**: The set from which we are selecting numbers is \( \{ 5^1, 5^2, 5^3, \ldots, 5^9 \} \). Let's denote these numbers as \( 5^1, 5^2, \ldots, 5^9 \).

2. **Understanding the Logarithm**: We want \( \log_a b \) to be an integer. By the properties of logarithms, we can express this as:
\[ \log_a b = \frac{\log b}{\log a} \]
Since \( a = 5^m \) and \( b = 5^n \) for some integers \( m \) and \( n \) (where \( 1 \leq m, n \leq 9 \)), we have:
\[ \log_a b = \frac{\log(5^n)}{\log(5^m)} = \frac{n \log 5}{m \log 5} = \frac{n}{m} \]
Thus, \( \log_a b \) is an integer if and only if \( \frac{n}{m} \) is an integer.

3. **Condition for Integer**: For \( \frac{n}{m} \) to be an integer, \( n \) must be a multiple of \( m \). This means \( n = km \) for some integer \( k \).

4. **Counting Valid Pairs**: We need to count the pairs \( (m, n) \) such that \( n \) is a multiple of \( m \) and both \( m \) and \( n \) are distinct and chosen from \( \{1, 2, \ldots, 9\} \).

- For \( m = 1 \): \( n \) can be \( 2, 3, 4, 5, 6, 7, 8, 9 \) (8 choices)
- For \( m = 2 \): \( n \) can be \( 4, 6, 8 \) (3 choices)
- For \( m = 3 \): \( n \) can be \( 6, 9 \) (2 choices)
- For \( m = 4 \): \( n \) can be \( 8 \) (1 choice)
- For \( m = 5, 6, 7, 8, 9 \): No valid \( n \) (0 choices)

Now, we sum the valid choices:
\[ 8 + 3 + 2 + 1 = 14 \]

5. **Total Distinct Pairs**: The total number of ways to choose 2 distinct numbers from 9 is given by \( \binom{9}{2} \):
\[ \binom{9}{2} = \frac{9 \times 8}{2} = 36 \]

6. **Calculating the Probability**: The probability that \( \log_a b \) is an integer is the number of valid pairs divided by the total pairs:
\[ P(\log_a b \text{ is an integer}) = \frac{14}{36} = \frac{7}{18} \]

### Final Answer:
The probability that \( \log_a b \) is an integer is \( \frac{7}{18} \).