If `log_(q)(xy)=3` and `log_(q)(x^(2)y^(3))=4`, find the value of `log_(q)x`,

Options

A. 4
B. 5
C. 3
D. 2

Answer

To solve the problem, we will use the properties of logarithms to find the value of \( \log_q x \).

### Step-by-Step Solution:

1. **Given Equations**:
We have two equations:
\[ \log_q (xy) = 3 \tag{1} \]
\[ \log_q (x^2 y^3) = 4 \tag{2} \]

2. **Using Logarithm Properties**:
From equation (1), we can use the property of logarithms that states \( \log_a (bc) = \log_a b + \log_a c \):
\[ \log_q x + \log_q y = 3 \tag{3} \]

3. **Expanding Equation (2)**:
For equation (2), we can also use the logarithm properties:
\[ \log_q (x^2 y^3) = \log_q (x^2) + \log_q (y^3) \]
Using the property \( \log_a (b^c) = c \cdot \log_a b \):
\[ 2 \log_q x + 3 \log_q y = 4 \tag{4} \]

4. **Substituting and Eliminating**:
Now we have two equations (3) and (4):
- From equation (3): \( \log_q y = 3 - \log_q x \)
- Substitute \( \log_q y \) in equation (4):
\[ 2 \log_q x + 3(3 - \log_q x) = 4 \]

5. **Simplifying the Equation**:
Expanding the equation:
\[ 2 \log_q x + 9 - 3 \log_q x = 4 \]
Combine like terms:
\[ -\log_q x + 9 = 4 \]
Rearranging gives:
\[ -\log_q x = 4 - 9 \]
\[ -\log_q x = -5 \]
Thus:
\[ \log_q x = 5 \]

### Final Answer:
The value of \( \log_q x \) is \( 5 \).