Let `a`, `b`, `c`, `d` are positive integer such that `log_(a)b=3//2` and `log_(c)d=5//4`. If `a-c=9`, then value of `(b-d)` is equal to

Options

A. `20`
B. `93`
C. `10`
D. `1`

Answer

To solve the problem step by step, we will use the properties of logarithms and the given conditions.

### Step 1: Rewrite the logarithmic equations
We are given:
1. \( \log_a b = \frac{3}{2} \)
2. \( \log_c d = \frac{5}{4} \)

Using the change of base formula, we can rewrite these equations as:
- From \( \log_a b = \frac{3}{2} \), we have:
\[ b = a^{\frac{3}{2}} = a^{1.5} \]
- From \( \log_c d = \frac{5}{4} \), we have:
\[ d = c^{\frac{5}{4}} = c^{1.25} \]

### Step 2: Use the given condition \( a - c = 9 \)
We also know that:
\[ a - c = 9 \]
This implies:
\[ a = c + 9 \]

### Step 3: Substitute \( a \) in terms of \( c \)
Substituting \( a \) into the expression for \( b \):
\[ b = (c + 9)^{\frac{3}{2}} \]

### Step 4: Substitute \( c \) in terms of \( d \)
From the expression for \( d \):
\[ d = c^{\frac{5}{4}} \]
We can express \( c \) in terms of \( d \):
\[ c = d^{\frac{4}{5}} \]

### Step 5: Substitute \( c \) back into the equation for \( b \)
Now substituting \( c \) into the equation for \( b \):
\[ b = \left(d^{\frac{4}{5}} + 9\right)^{\frac{3}{2}} \]

### Step 6: Find \( b - d \)
Now we want to find \( b - d \):
\[ b - d = \left(d^{\frac{4}{5}} + 9\right)^{\frac{3}{2}} - d \]

### Step 7: Solve for specific values
To find specific values, we can try some integers for \( d \) and see if we can satisfy the equations.

Assuming \( d = 32 \):
1. Calculate \( c \):
\[ c = 32^{\frac{4}{5}} = 16 \]
2. Calculate \( a \):
\[ a = c + 9 = 16 + 9 = 25 \]
3. Calculate \( b \):
\[ b = (25)^{\frac{3}{2}} = 125 \]

### Step 8: Calculate \( b - d \)
Now we can find:
\[ b - d = 125 - 32 = 93 \]

### Final Answer
Thus, the value of \( b - d \) is:
\[ \boxed{93} \]