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**1. 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.

**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'.

** Characteristic: **The internal part of the logarithm of a number is called its

**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:-**

*Mantissa:*

The decimal part of the logarithm of a number is known is its *mantissa.* For mantissa, welook 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}

**Answer. A**

**Explanation.**

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"

Choice A is the correct answer.

**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

**Answer. D**

**Explanation.**

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."

Choice D is the correct answer.

**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

**Answer. C**

**Explanation.**

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?"

Choice C is the correct answer.

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

A.

B.

C.

D.

**Answer. D**

**Explanation.**

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."

Choice D is the correct answer.

**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

**Answer. A**

**Explanation.**

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."

Choice A is the correct answer.

**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.

**Answer. A**

**Explanation.**

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?"

Choice A is the correct answer.

**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}

**Answer. C**

**Explanation.**

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."

Choice A is the correct answer.

**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}

**Answer. B**

**Explanation.**

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}."

Choice B is the correct answer.

**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

**Answer. B**

**Explanation.**

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?"

Choice B is the correct answer.

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

A. 3

B. 6

C. 7

D. 11

**Answer. D**

**Explanation.**

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?"**

Choice D is the correct answer.

**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

**Answer. A**

**Explanation**

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."**

Choice A is the correct answer.

**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

**Answer. D**

**Explanation.**

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."**

Choice D is the correct answer.

**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

**Answer. A**

**Explanation.**

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?"**

Choice A is the correct answer.

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