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
2. Power rule of logarithm
3. Product rule of logarithm 3 concepts
If log 2, log (2^{x} 1) and log (2^{x} + 3) are in A.P, then x is equal to ____
 \\frac {5}{2} \\)
 log_{2}5
 log_{3}2
 \\frac {3}{2} \\)
Correct Answer Choice (2). The value of x is log_{2}5.
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
 Product Rule: log a + log b = log ab.
Example: log 2 + log 10 = log 20  Power Rule:a log b = log b^{a}
Example: 3 log 2 = log 2^{3} = log 8
 Product Rule: log a + log b = log ab.

Explanatory Answer
Because log 2, log (2^{x} 1) and log (2^{x} + 3) are in an AP
2 log (2^{x}  1) = log 2 + log (2^{x} + 3)Using the power rule of log, we can write 2 log (2^{x}  1) = log (2^{x}  1)^{2}
Therefore, we can write the equation as log (2^{x}  1)^{2} = log 2 + log (2^{x} + 3)
Using product rule, we can write the right hand side of the equation as follows:
log 2 + log (2^{x} + 3) = log 2 (2^{x} + 3)The modified equation after applying the power rule and the product rule of logarthims is log (2^{x}  1)^{2} = log 2 (2^{x} + 3)
Removing logarithm on both sides of the equation, we get
(2^{x}  1)^{2} = 2 (2^{x} + 3)
Or 2^{2x}  2*2^{x} + 1 = 2 * 2^{x} + 6
Or 2^{2x}  4 * 2^{x}  5 = 0Let y = 2^{x}
So, the equation can be written as y^{2}  4y  5 = 0
Factorzing, we get (y  5)(y + 1) = 0.
So, y = 5 or y = 1So, 2^{x} = 5 or 2^{x} = 1
2^{x} cannot be negative. S, 2^{x} = 5If 2^{x} = 5, expressed in terms of a log, it becomes x = log_{2}5
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 a_{n} = a_{1} + (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 =
Alternatively, you can find the sum using this formula S_{n} =
More Questions on Arithmetic, Geometric Progression
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 TANCET 2013: Sum of a GP
 XAT 2014: AP & Divisibility
 XAT 2015: Sum of an AP
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