The document Logarithms - Important Formulas, Quantitative Aptitude GMAT Notes | EduRev is a part of the CAT Course Quantitative Aptitude for GMAT.

All you need of CAT at this link: CAT

**Logarithm**

- If a is a positive real number, other than 1 and a
^{m}= x, then we write: m = log_{a}x and we say that the value of log x to the base a is m. **Examples:****(i)**10^{3}1000 ⇒ log_{10}1000 = 3.**(ii)**3^{4}= 81 ⇒ log_{3}81 = 4.**(iv)**(. 1)^{2}= 01 ⇒ log_{(.1)}.01 = 2.

**Properties of Logarithms**

1. log a (xy) = log_{a} x + log_{a} y

3. log_{x} x = 1

4. log_{a} 1 = 0

5. log_{a} (x^{n}) = n(log_{a} 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:**

*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 log _{2}X + log_{4}X = log_{0.25}√6 and x > 0, then x is:**

**A. **6^{-1/6}

**B.** 6^{1/6}

**C.** 3^{-1/3}

**D.** 6^{1/3}

**Correct Answer is Option (A).**

- log
_{2}x + log_{4}x = log_{0.25}√6

We can rewrite the equation as:

⇒ log_{2}x * 3 = 2log_{0.25}√6

⇒ log_{2}x^{3}= -log_{4}6

⇒

⇒

⇒ 2log_{2}x^{3}= -log_{2}6 - 2log
_{2}x^{3}+ log_{2}6 = 0

log_{2}6X^{6}= 0 - 6x
^{6}= 1

x^{6}= 1/6 - The question is "If log
_{2}X + log_{4}X = log_{0.25}√6 and x > 0, then x is" - Hence, the answer is "6
^{-1/6}".

**Question 2****:** log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x))) = 1/2. Find x.

A. 4

B. 1/2

C. 1

D. 2

**Correct Answer is Option (D).**

log_{9} (3log_{2} (1 + log_{3} (1 + 2log_{2}x)) = 1/2

3log_{2}(1 + log_{3}(1 + 2log_{2}x)) = 9^{1/2} = 3

log_{2}(1 + log_{3}(1 + 2log_{2}x) = 1

1 + log_{3}(1 + 2log_{2}x) = 2

log_{3}(1 + 2log_{2}x) = 1

1 + 2log_{2}x = 3

2log_{2}x = 2

log_{2}x = 1

x = 2

The question is "Find x."

**Question 3****:** If 2^{2x+4} – 17 × 2^{x+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).**

2^{x+4} – 17 * 2^{x+1} = – 4

=> 2^{x+1} = y

2^{2x+2} = y^{2}

2^{2}(2^{2x+2}) – 17 * 2^{x+1} = –4

4y^{2} – 17y + 4 = 0

4y^{2} – 16y – y + y = 0

4y (y – 4) – 1 (y – 4) = 0

y = 1/4 or 4

2^{x+1} = 1/4 or 4

⇒ x + 1 = 2 or – 2

x = 1 or – 3

The question is "which of the following is true?"

**Question 4****:** If log_{12}27 = a, log_{9}16 = b, find log_{8}108

A.

B.

C.

D.

**Correct Answer is Option (D).**

log_{8}108 = log_{8}(4 * 27)

log_{8}108 = log_{8}4 + log_{8}27

⇒ log_{8}4 = 2/3

log_{8}27 = 2 * log_{16}9

log_{9}16 = b

log_{16}9 = 1/b

log_{8}27 = 2/b

The question is "find log_{8}108."

**Question 5****:** If a, b are integers such that x = a, and x = b satisfy this inequation, find the maximum possible value of a – b.

A. 214

B. 216

C. 200

D. 203

**Correct Answer is Option (A).**

log_{3}x = y

⇒

y ∈ (3, 5)

3 < log_{3}x < 5

27 < x < 243

Therefore max ( a – b) will be when a = 242 and b = 28. Therefore, max(a – b) = 214.

The question is "find the maximum possible value of a – b."

**Question 6****:** log_{5}x = a (This should be read as log X to the base 5 equals a) log_{20}x = b. What is log_{x}10?

A.

B. (a + b) * 2ab

C.

D.

**Correct Answer is Option (A).**

Given, log_{5}x = a

log_{20}x = b

log_{x}5 = 1/a

log_{x}20 = 1/b

⇒

The question is "What is log_{x}10?"

**Question 7**: log_{3}x + log_{x}3 = 17/4. Find x.

A. 34

B. 3^{1/8}

C. 3^{1/4}

D. 3^{1/3}

**Correct Answer is Option (C).**

log_{3}x + log_{x}3 = 17/4

Let y = log_{3}x

We know that log_{x}3 =

Hence log_{x}3 = 1/y

Thus the equation can be written as

4y^{2} + 4 = 17y

4y^{2} + 4 - 17y = 0

Solving the above equation we get y = 4 or 1/4

If y = 4

log_{3}x = 4

Then x = 3^{4}

If y = 1/4

log_{3}x = 1/4

Then x = 3^{1/4}

The question is "Find x."

**Question 8****:** log_{x}y + log_{y}x^{2} = 3. Find log_{x}y^{3}.

A. 4

B. 3

C. 3^{1/2}

D. 3^{1/16}

**Correct Answer is Option (B).**

log_{x}y + log_{y}x^{2} = 3

Let a = log_{x}y

log_{y}x^{2} = 2log_{y}x

We know that log_{y}x =

Hence form above log_{y}x = 1/a

Now rewritting the equation log_{x}y + log_{y}x^{2} = 3

Using a we get

i.e., a^{2} - 3a + 2 = 0

Solving we get a = 2 or 1

If a = 2, Then log_{x}y = 2 and log_{y}x^{3} = 3

log_{x}y = 3 * 2 = 6

Or

If a = 1, Then log_{x}y = 1 and log_{y}x^{3} = 3

log_{x}y = 3 * 1 = 3

The question is "Find log_{x}y^{3}."

**Question 9****:** log_{2} 4 * log_{4} 8 * log_{8} 16 * ……………nth term = 49, what is the value of n?

A. 49

B. 48

C. 34

D. 24

**Correct Answer is Option (B).**

First, the nth term of L.H.S need to be defined by observing the pattern :-

It is log_{(2n)} 2.2^{n}

Given,

log_{2} 4 * log_{4} 8 * log_{8} 16 * ……………log_{(2n)} 2.2n = 49

Whenever solving a logarithm equation, generally one should approach towards making the base same.

Making the base 2:-

log_{(2n)} 2.2^{n} = 49

log_{(2n)} 2 + log_{(2n)} 2^{n} = 49

1 + n = 49

n = 48

The question is "what is the value of n?"

**Question 10****:** If 3^{3 + 6 + 9 + ……… 3x }= what is the value of x?

A. 3

B. 6

C. 7

D. 11

**Correct Answer is Option (D).**

First of all, let us define the x^{th} term.

Whenever you encounter a distinctive number such as one given in R.H.S of above equation, always try to find its significance in the context of question.

In this case L.H.S has 3^{a}, so must be some form of 3^{a}.

With little hit and trial, you may find

3^{3(1 + 2 + 3 + ...X)} = 3^{ -3 * -66}

⇒ 3^{3} * 3^{x(x+1)/2} = 3^{3*66}

x(x+1) = 132

Solving this equation for x > 0, we get x = 11.

You should directly be able to see that 132 = 11 * 12 => x= 11

And avoid wasting time solving the complete equation.

The question is **"what is the value of x?"**

**Question 11****:** x, y, z are 3 integers in a geometric sequence such that y - x is a perfect cube. Given, log_{36}x^{2} + log_{6}√y + log_{216}y^{1/2}z = 6. Find the value of x + y + z.

A. 189

B. 190

C. 199

D. 201

**Correct Answer is Option (A).**

Let us begin with simplifying the equation:-

⇒ log_{62}x^{2} + log_{6}y^{1/2} + 3log_{63}y^{1/2}z = 6

log_{6}x + log_{6}y^{1/2}y^{1/2}z = 6

log_{6}xyz = 6

xyz = 6^{6}

Given x,y,z is in G.P. Let x = a, y = ab, z = ab^{2}

⇒ xyz = a^{3}b^{3} = (ab)^{3}

(ab)^{3} = (6^{2})^{3}

Possible values of (a,b) satisfying the equation :-

(1, 36), (2, 18), (3, 12), (4, 9), (9, 4), (12, 3), (18, 2), (36, 1)

Given y-x is a perfect cube

⇒ ab-a is perfect cube

⇒ a(b-1) is perfect cube

Only possible when (a, b) = (9, 4)

∴ x = 9 , y = 36 , z = 144

∴ x + y + z = 9 + 36 + 144 = 189

The question is **"Find the value of x + y + z."**

**Question 12****: **10^{log(3 - 10logy)} = log_{2}(9 - 2^{y}), Solve for y.

A. 0

B. 3

C. 0 and 3

D. none of these

**Correct Answer is Option (D).**

Before beginning to simplify the equation, don’t forget that anything inside a log cannot be negative

10^{log(3-y)} = log_{2}(9 - 2^{y}) (y > 0)…………………………………(1)

3 - y = log_{2}(9 - 2^{y}) (Therefore, 3 - y > 0 =) (y < 3)) ……………………… (2)

2^{3-y} = 9 - 2^{y}

2^{y} = t

⇒ 8 = 9t –t^{2}

⇒ t^{2} - 9t + 8 = 0

⇒ t^{2} - t - 8t - 8 = 0

⇒ t(t - 1) - 8(t - 1) = 0

⇒ t = 1, 8

Therefore, 2^{y} = 1 and 2^{y} = 8

⇒ y = 0 and y = 3

However, from inequalities (1) and (2), y cannot take either of these value.

The question is **"Solve for y."**

**Question 13****:** 4^{6+12+18+24+…+6x} = (0.0625)^{-84}, what is the value of x?

A. 7

B. 6

C. 9

D. 12

**Correct Answer is Option (A).**

Take right side expression,

= (4^{-2})^{-84}

= 4^{168}

Take left side expression

4^{6+12+18+24+…+6x} = 4^{6(1+2+3+4+x)}

= 4^{6} * 4^{(1+2+3+4+…+x)}

= 4^{6} * 4^{x(x+1)/2} (using the formula for sum of natural numbers from 1 to x)

Equating left and right side expresssions, we get 4^{6} * 4^{x(x+1)/2} = 4^{168}

Or 4^{6} * 4^{x(x+1)/2} = 4^{6*28}

⇒

or x (x + 1) = 56

Solving for x we get, x = 7

The question is **"what is the value of x?"**

Offer running on EduRev: __Apply code STAYHOME200__ to get INR 200 off on our premium plan EduRev Infinity!

80 videos|99 docs|175 tests