The number of solutions of `log_(x)3. = 2x-3` is,

Options

A. 3
B. 1
C. 2
D. 0

Answer

To solve the equation \( \log_x 3 = 2x - 3 \), we will follow these steps:

### Step 1: Rewrite the logarithmic equation
The equation can be rewritten using the change of base formula:
\[ \log_x 3 = \frac{\log 3}{\log x} \]
Thus, we can rewrite the equation as:
\[ \frac{\log 3}{\log x} = 2x - 3 \]

### Step 2: Define functions for graphical analysis
Let:
- \( y_1 = \frac{\log 3}{\log x} \) (LHS)
- \( y_2 = 2x - 3 \) (RHS)

### Step 3: Analyze the function \( y_1 \)
The function \( y_1 = \frac{\log 3}{\log x} \) is defined for \( x > 0 \) and \( x \neq 1 \):
- As \( x \to 0^+ \), \( \log x \to -\infty \) and \( y_1 \to 0 \).
- At \( x = 1 \), \( \log x = 0 \) which makes \( y_1 \) undefined.
- As \( x \to \infty \), \( \log x \to \infty \) and \( y_1 \to 0 \).

### Step 4: Analyze the function \( y_2 \)
The function \( y_2 = 2x - 3 \) is a linear function:
- It intersects the y-axis at \( y = -3 \) when \( x = 0 \).
- It has a slope of 2, meaning it increases steadily.

### Step 5: Find points of intersection
To find the number of solutions, we need to determine how many times the graphs of \( y_1 \) and \( y_2 \) intersect:
- \( y_1 \) approaches 0 from above as \( x \) approaches 1 from the left and from below as \( x \) approaches 1 from the right.
- \( y_2 \) is a straight line that continues to rise.

### Step 6: Sketch the graphs
1. **Graph of \( y_1 \)**: Starts from \( (0, \infty) \), approaches \( (1, \infty) \) (undefined), and then decreases towards \( (1, 0) \).
2. **Graph of \( y_2 \)**: Starts at \( (0, -3) \) and moves upwards.

### Step 7: Determine intersections
- The graph of \( y_1 \) will intersect \( y_2 \) twice:
- Once when \( x < 1 \) (as \( y_1 \) decreases from \( \infty \) to \( 0 \)).
- Once when \( x > 1 \) (as \( y_1 \) increases from \( 0 \) to \( \infty \)).

### Conclusion
Thus, the number of solutions to the equation \( \log_x 3 = 2x - 3 \) is:
\[ \boxed{2} \]