If equations Vloga bloga √b and a-loga b = loga (ab) satisfy for all real values, then find the value of b4.

(A) (1.25)4

(B) (1.25)16

(C) (1.25)32

(D) (1.25)64

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: 

\[
\boxed{B: (1.25)^{64}}
\]

Explore how to solve equations involving logarithms and exponents with a detailed step-by-step solution. This problem covers key concepts such as logarithmic properties and exponential equations. Suitable for high school students.

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