GMAT Exam > GMAT Notes > Quantitative for GMAT > Important Formulas: Arithmetic Progression
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Logarithm
- If a is a positive real number, other than 1 and am = x, then we write: m = loga x and we say that the value of log x to the base a is m.
- Examples: (i) 103 1000 ⇒ log10 1000 = 3.
(ii) 34 = 81 ⇒ log3 81 = 4.
(iv) (. 1)2 = 01 ⇒ log(.1) .01 = 2.
Properties of Logarithms
1. log a (xy) = loga x + loga y
3. logx x = 1
4. loga 1 = 0
5. loga (xn) = n(loga x)
Common Logarithms
- Logarithms to the base 10 are known as common logarithms.
- The logarithm of a number contains two parts, namely 'characteristic' and 'mantissa'.
1. Characteristic
The internal part of the logarithm of a number is called its characteristic.
Case I: When the number is greater than 1.
- In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.
Case II: When the number is less than 1.
- In this case, the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and it is negative.
- Instead of -1, -2 etc. we write (two bar), etc.
Examples:
(two bar), etc.
2. Mantissa:
The decimal part of the logarithm of a number is known is its mantissa. For mantissa, we look through log table.
Question 1: If log2X + log4X = log0.25√6 and x > 0, then x is:
A. 6-1/6
B. 61/6
C. 3-1/3
D. 61/3
Correct Answer is Option (A).
- log2x + log4x = log0.25√6
We can rewrite the equation as:
⇒ log2x * 3 = 2log0.25√6
⇒ log2x3 = -log46
⇒⇒
⇒ 2log2x3 = -log26 - 2log2x3 + log26 = 0
log26X6 = 0 - 6x6 = 1
x6 = 1/6 - The question is "If log2X + log4X = log0.25 √6 and x > 0, then x is"
- Hence, the answer is "6-1/6".
⇒
Question 2: log9 (3log2 (1 + log3 (1 + 2log2x))) = 1/2. Find x.
A. 4
B. 1/2
C. 1
D. 2
Correct Answer is Option (D).
log9 (3log2 (1 + log3 (1 + 2log2x)) = 1/2
3log2(1 + log3(1 + 2log2x)) = 91/2 = 3
log2(1 + log3(1 + 2log2x) = 1
1 + log3(1 + 2log2x) = 2
log3(1 + 2log2x) = 1
1 + 2log2x = 3
2log2x = 2
log2x = 1
x = 2
The question is "Find x."
Hence, the answer is "2".
Question 3: If 22x+4 – 17 × 2x+1 = –4, then which of the following is true?
A. x is a positive value
B. x is a negative value
C. x can be either a positive value or a negative value
D. None of these
Correct Answer is Option (C).
2x+4 – 17 * 2x+1 = – 4
=> 2x+1 = y
22x+2 = y2
22(22x+2) – 17 * 2x+1 = –4
4y2 – 17y + 4 = 0
4y2 – 16y – y + y = 0
4y (y – 4) – 1 (y – 4) = 0
y = 1/4 or 4
2x+1 = 1/4 or 4
⇒ x + 1 = 2 or – 2
x = 1 or – 3
The question is "which of the following is true?"
Hence, the answer is "x can be either a positive value or a negative value".
Logarithmic Series
Definition
An expansion for loge (1 + x) as a series of powers of x which is valid only, when |x|<1.
Expansion of logarithmic series
Expansion of loge (1 + x) if |x|<1 then
Replacing x by −x in the logarithmic series, we get
Some Important results from logarithmic series
(1) 
(2) The series expansion of loge (1 + x) may fail to be valid, if |x| is not less than 1. It can be proved that the logarithmic series is valid for x = 1. Putting x = 1 in the logarithmic series.
We get,
(3) When x = −1, the logarithmic series does not have a sum. This is in conformity with the fact that log(1 – 1) is not a finite quantity.
Difference between the exponential and logarithmic series
- In the exponential series all the terms carry positive signs whereas in the logarithmic series the terms are alternatively positive and negative
- In the exponential series the denominator of the terms involve factorial of natural numbers. But in the logarithmic series the terms do not contain factorials.
- The exponential series is valid for all the values of x. The logarithmic series is valid, when |x|< 1.
all the terms carry positive signs whereas in the logarithmic series 
the terms are alternatively positive and negative
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FAQs on Important Formulas: Arithmetic Progression - Quantitative for GMAT
| 1. What are the properties of logarithms? |  |
Ans. The properties of logarithms are as follows:
- Product Rule: logₐ(xy) = logₐ(x) + logₐ(y)
- Quotient Rule: logₐ(x/y) = logₐ(x) - logₐ(y)
- Power Rule: logₐ(xⁿ) = n * logₐ(x)
- Change of Base Rule: logₐ(x) = logᵦ(x) / logᵦ(a)
- Inverse Rule: If logₐ(x) = y, then aⁿ = x
| 2. What are the important formulas related to arithmetic progression? | |
Ans. The important formulas related to arithmetic progression (AP) are:
- nth term of an AP: aₙ = a + (n-1)d
- Sum of first n terms of an AP: Sₙ = (n/2)(2a + (n-1)d)
- Sum of first n terms of an AP using the last term: Sₙ = (n/2)(a + l)
- Sum of squares of first n natural numbers: Sₙ = (n/6)(n+1)(2n+1)
| 3. How do you use the product rule of logarithms? | |
Ans. To use the product rule of logarithms, you can split a product inside a logarithm into separate logarithms. For example, if you have logₐ(xy), you can rewrite it as logₐ(x) + logₐ(y). This allows you to simplify the calculation by evaluating the logarithms of the individual factors separately and then adding them.
| 4. How do you find the nth term of an arithmetic progression? | |
Ans. To find the nth term (aₙ) of an arithmetic progression (AP), you can use the formula aₙ = a + (n-1)d, where a is the first term and d is the common difference between consecutive terms. By substituting the values of a and d into the formula, you can calculate the desired term of the AP.
| 5. How do you find the sum of the first n terms of an arithmetic progression? | |
Ans. To find the sum (Sₙ) of the first n terms of an arithmetic progression (AP), you can use the formula Sₙ = (n/2)(2a + (n-1)d), where a is the first term, d is the common difference, and n is the number of terms. By plugging in the values of a, d, and n into the formula, you can calculate the sum of the desired terms.
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