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AP and Logarithm Practice Question

An interesting arithmetic progression problem where the terms are log of expressions in x. You need to know the basics of AP and rules log to solve this question. In the explanatory answer section, we have provided the theoretical framework in the first tab and the complete solution in the second tab. You are likely to see such a question only in the CAT or XAT.

Question

If log 2, log (2x -1) and log (2x + 3) are in A.P, then x is equal to ____

  1. \\frac {5}{2} \\)
  2. log25
  3. log32
  4. \\frac {3}{2} \\)

  Correct Answer      Choice (2). The value of x is log25.


Explanatory Answer

  • Concepts in Arithmetic Progression

    If three numbers a, b, and c are in Arithmetic progression it means b - a = c - b
    Or, we can rewrite it as 2b = a + c.

    Concepts in Logarithm

    1. Product Rule: log a + log b = log ab.
      Example: log 2 + log 10 = log 20

    2. Power Rule:a log b = log ba
      Example: 3 log 2 = log 23 = log 8
  • 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 = log25

      The correct answer is Choice (2).

  • Formulae to remember in Arithmetic Sequence

    An arithmetic progression is such a sequence in which every subsequent term of the sequence is obtained by adding the preceding term with a constant number. This constant number is called the common difference(d) of the sequence.

    e.g., 2, 4, 6, 8, 10 is an arithmetic sequence.

    The second term is obtained by adding the first term by 2. The third term is obtained by adding the second term by 2 and so on.

    Formula 1: nth term of a Arithmetic Progression an = a1 + (n - 1)d, where a is the first term of the sequence and d is the common difference.

    Formula 2: Sum of the first n terms of a arithmetic progression = (n/2)(a1 + an)

    Alternatively, you can find the sum using this formula Sn = (n/2)(2a1 + (n-1)d)

More Questions on Arithmetic, Geometric Progression

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