If log 2, log (2x -1) and log (2x + 3) are in A.P, then x is equal to ____
Pick
A) 5/2
B) log(2)5
C) log(3)2
D) 3/2
E) None of these
Explanatory Answer   Because log 2, log (2x -1) and log (2x + 3) are in an AP   2 log (2x - 1) = log 2 + log (2x + 3)   Using the power rule of log, we can write 2 log (2x - 1) = log (2x - 1)2   Therefore, we can write the equation as log (2x - 1)2 = log 2 + log (2x + 3)   Using product rule, we can write the right hand side of the equation as follows:   log 2 + log (2x + 3) = log 2 (2x + 3)   The modified equation after applying the power rule and the product rule of logarthims is log (2x - 1)2 = log 2 (2x + 3)   Removing logarithm on both sides of the equation, we get   (2x - 1)2 = 2 (2x + 3)   Or 22x - 2*2x + 1 = 2 * 2x + 6   Or 22x - 4 * 2x - 5 = 0   Let y = 2x   So, the equation can be written as y2 - 4y - 5 = 0   Factorzing, we get (y - 5)(y + 1) = 0.   So, y = 5 or y = -1   So, 2x = 5 or 2x = -1   2x cannot be negative. S, 2x = 5   If 2x = 5, expressed in terms of a log, it becomes x = log(2)5   Answer log(2)5
<result>B</result>
If log 2, log (2x -1) and log (2x + 3) are in A.P, then x is equal to ____