If n is a natural number such that `n=P_1^(a_1)P_2^(a_2)P_3^(a_3)...P_k^(a_k)` where `p_1,p_2,...p_k` are distinct primes then minimum value of `log n` is:

Options

A. `k "log" 2`
B. `k "log" 3`
C. `k "log" 4`
D. none of these

Explanation

To solve the problem, we need to find the minimum value of \( \log n \) given that \( n \) can be expressed as a product of distinct prime numbers raised to their respective powers. Let's break it down step by step.

### Step 1: Understand the expression for \( n \)
We are given that:
\[ n = P_1^{a_1} \times P_2^{a_2} \times P_3^{a_3} \times \ldots \times P_k^{a_k} \]
where \( P_1, P_2, \ldots, P_k \) are distinct prime numbers and \( a_1, a_2, \ldots, a_k \) are their respective positive integer powers.

### Step 2: Identify the minimum value of \( n \)
To find the minimum value of \( n \), we should consider the smallest prime number, which is 2. If we take all \( P_i \) to be 2, then:
\[ n \geq 2^{a_1 + a_2 + a_3 + \ldots + a_k} \]
If we assume \( a_1 = a_2 = a_3 = \ldots = a_k = 1 \) (the smallest possible values for each exponent), we get:
\[ n \geq 2^k \]

### Step 3: Taking the logarithm
Now, we take the logarithm of both sides:
\[ \log n \geq \log(2^k) \]
Using the logarithmic identity \( \log(a^b) = b \log a \), we can simplify this to:
\[ \log n \geq k \log 2 \]

### Step 4: Conclusion
Thus, the minimum value of \( \log n \) is:
\[ \log n \geq k \log 2 \]
This means the minimum value of \( \log n \) is \( k \log 2 \).

### Final Answer
The minimum value of \( \log n \) is:
\[ \boxed{k \log 2} \]