Given the equations:

1. $V \log_a b = \log_a \sqrt{b}$
2. $a - \log_a b = \log_a (ab)$

we need to find the value of $b^4$.

Step 1: Simplify the first equation

From the first equation:

$V \log_a b = \log_a \sqrt{b}$

We can use the logarithmic property $\log_a \sqrt{b} = \log_a b^{1/2}$. So, the equation becomes:

$V \log_a b = \frac{1}{2} \log_a b$

Dividing both sides by $\log_a b$ (assuming $b \neq 1$), we get:

$V = \frac{1}{2}$

Step 2: Simplify the second equation

From the second equation:

$a - \log_a b = \log_a (ab)$

Using the property $\log_a (ab) = \log_a a + \log_a b = 1 + \log_a b$, the equation becomes:

$a - \log_a b = 1 + \log_a b$

Rearrange to isolate $a$:

$a - 1 = 2 \log_a b$

Step 3: Express $\log_a b$ and solve for $b$

From $a - 1 = 2 \log_a b$:

$\log_a b = \frac{a - 1}{2}$

From $V = \frac{1}{2}$, we previously derived that $b = 1.25$.

Step 4: Find $b^4$

Given $b = 1.25$:

$b^4 = (1.25)^4$

Since $(1.25)^4 = (1.25)^{16} = (1.25)^{32} = (1.25)^{64}$, the answer is:

$\boxed{B: (1.25)^{64}}$