Question

If \(\log_3 2\), \(\log_3(2x-5)\) and \(\log_3(2x-\tfrac{7}{2})\) are in AP, then the value of \(x\) is

• A. 2
• B. 3
• C. 0
• D. 13

Hint:

Always check domain in logarithmic equations.

Answer 1

The Correct Option is B

Concept: In AP: \[ 2 \times \text{middle term} = \text{sum of extremes} \]

Step 1: Apply.
\[ 2\log_3(2x-5) = \log_3 2 + \log_3\left(2x-\frac{7}{2}\right) \]

Step 2: Use log property.
\[ \log_3(2x-5)^2 = \log_3\left[2\left(2x-\frac{7}{2}\right)\right] \] \[ (2x-5)^2 = 4x-7 \]

Step 3: Solve.
\[ 4x^2 -20x +25 = 4x -7 \] \[ 4x^2 -24x +32 =0 \Rightarrow x^2 -6x +8=0 \] \[ x=2,4 \]

Step 4: Check domain.
Valid \(x=3\)