The minimum value of 'c' such that `log_(b)(a^(log_(2)b))=log_(a)(b^(log_(2)b)) and log_(a) (c-(b-a)^(2))=3`, where `a, b in N` is :

Answer

To solve the problem step by step, we will break down the equations given and find the minimum value of 'c'.

### Step 1: Analyze the given equation
We start with the equation:
\[ \log_b(a^{\log_2 b}) = \log_a(b^{\log_2 b}) \]

### Step 2: Apply the logarithmic property
Using the property of logarithms that states \(\log_y(x^a) = a \cdot \log_y(x)\), we can rewrite both sides:
\[ \log_2 b \cdot \log_b a = \log_2 b \cdot \log_a b \]

### Step 3: Simplify the equation
Since \(\log_2 b\) is common on both sides, we can divide by \(\log_2 b\) (assuming \(b \neq 1\)):
\[ \log_b a = \log_a b \]

### Step 4: Use the property of logarithms
The equation \(\log_b a = \log_a b\) implies:
\[ \frac{1}{\log_a b} = \log_a b \]
Let \(x = \log_a b\). Then, we have:
\[ x^2 = 1 \implies x = 1 \text{ (since } a, b \in \mathbb{N} \text{ and } a \neq b \text{)} \]

### Step 5: Conclude the relationship between a and b
From \(x = 1\), we conclude:
\[ \log_a b = 1 \implies a = b \]

### Step 6: Substitute into the second equation
Now, we substitute \(a = b\) into the second equation:
\[ \log_a(c - (b - a)^2) = 3 \]
Since \(b - a = 0\), we have:
\[ \log_a(c - 0^2) = 3 \implies \log_a(c) = 3 \]

### Step 7: Solve for c
Using the property of logarithms:
\[ c = a^3 \]

### Step 8: Find the minimum value of c
Since \(a\) and \(b\) are natural numbers and \(a = b\), the smallest natural number \(a\) can take is 2 (since \(a\) cannot be 1):
\[ c = 2^3 = 8 \]

### Conclusion
Thus, the minimum value of \(c\) is:
\[ \boxed{8} \]