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 (2x -1) and log (2x + 3) are in A.P, then x is equal to ____
- \\frac {5}{2} \\)
- log25
- log32
- \\frac {3}{2} \\)
Correct Answer Choice (2). The value of x is log25.
Explanatory Answer
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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 ba
Example: 3 log 2 = log 23 = log 8
- Product Rule: log a + log b = log ab.
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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 = 0Let 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 = -1So, 2x = 5 or 2x = -1
2x cannot be negative. S, 2x = 5If 2x = 5, expressed in terms of a log, it becomes x = log25
The correct answer is Choice (2).
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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 =
Alternatively, you can find the sum using this formula Sn =
More Questions on Arithmetic, Geometric Progression
- Find the sum of terms of a GP
- AP: find a term; sum known
- Sum of AP having -ve common difference
- AP: find a terms; sum & product given
- Sum of an AP Series
- CAT 2003: Find sum of terms of an AP
- AP: Integers divisible by 4 or 9
- TANCET 2013: Sum of a GP
- XAT 2014: AP & Divisibility
- XAT 2015: Sum of an AP
- Logarithm terms in an AP
- Speed Distance Time & AP
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