If `log_(3)2,log_(3)(2^(x)-5)` and `log_(3)(2^(x)-7/2)` are in A.P., then x is equal to

Options

A. 2
B. 3
C. 4
D. 2.3

Answer

To solve the problem where \( \log_3 2 \), \( \log_3 (2^x - 5) \), and \( \log_3 \left(2^x - \frac{7}{2}\right) \) are in Arithmetic Progression (A.P.), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding A.P. Condition**:
If three numbers \( a, b, c \) are in A.P., then the condition is:
\[ 2b = a + c \]
Here, let:
- \( a = \log_3 2 \)
- \( b = \log_3 (2^x - 5) \)
- \( c = \log_3 \left(2^x - \frac{7}{2}\right) \)

2. **Applying A.P. Condition**:
Substitute \( a, b, c \) into the A.P. condition:
\[ 2 \log_3 (2^x - 5) = \log_3 2 + \log_3 \left(2^x - \frac{7}{2}\right) \]

3. **Using Logarithmic Properties**:
Recall the properties of logarithms:
- \( \log_b a + \log_b c = \log_b (a \cdot c) \)
- \( \log_b (a^m) = m \cdot \log_b a \)

Therefore, we can rewrite the equation:
\[ 2 \log_3 (2^x - 5) = \log_3 \left(2 \cdot \left(2^x - \frac{7}{2}\right)\right) \]

4. **Simplifying the Equation**:
Using the property \( 2 \log_3 (2^x - 5) = \log_3 ((2^x - 5)^2) \), we rewrite the equation:
\[ \log_3 ((2^x - 5)^2) = \log_3 \left(2 \cdot \left(2^x - \frac{7}{2}\right)\right) \]

5. **Removing the Logarithm**:
Since the logarithms are equal, we can equate the arguments:
\[ (2^x - 5)^2 = 2 \left(2^x - \frac{7}{2}\right) \]

6. **Expanding Both Sides**:
Expanding the left side:
\[ (2^x - 5)^2 = 2^{2x} - 10 \cdot 2^x + 25 \]
Expanding the right side:
\[ 2 \left(2^x - \frac{7}{2}\right) = 2^{x+1} - 7 \]

7. **Setting Up the Equation**:
Now we have:
\[ 2^{2x} - 10 \cdot 2^x + 25 = 2^{x+1} - 7 \]
Rearranging gives:
\[ 2^{2x} - 10 \cdot 2^x - 2^{x+1} + 32 = 0 \]

8. **Substituting \( t = 2^x \)**:
Let \( t = 2^x \), then the equation becomes:
\[ t^2 - 10t - 2t + 32 = 0 \]
Simplifying gives:
\[ t^2 - 12t + 32 = 0 \]

9. **Factoring the Quadratic**:
Factor the quadratic:
\[ (t - 8)(t - 4) = 0 \]
Therefore, \( t = 8 \) or \( t = 4 \).

10. **Finding \( x \)**:
Recall \( t = 2^x \):
- If \( t = 8 \), then \( 2^x = 2^3 \) implies \( x = 3 \).
- If \( t = 4 \), then \( 2^x = 2^2 \) implies \( x = 2 \).

11. **Checking Validity**:
Check both values of \( x \):
- For \( x = 2 \): \( \log_3 (2^2 - 5) = \log_3 (-1) \) (not valid).
- For \( x = 3 \): All logarithmic values are valid.

Thus, the final answer is:
\[ \boxed{3} \]