LOGARITHMS

the logarithm of a number is the exponent to which another fixed value, the base, must be raised to produce that number. For example, the logarithm of 1000 to base 10 is 3, because 10 to the power 3 is 1000: 1000 = 10 × 10 × 10 = 10³.The logarithm to base 10 (b = 10) is called the common logarithm and has many applications in science and engineering. The natural logarithm has the irrational (transcendental) number e (≈ 2.718) as its base; its use is widespread in pure mathematics, especially calculus. The binary logarithm uses base 2 (b = 2) and is prominent in computer science.

Some properties of logarithms are given below:
 If a^m = x, then m = loga(x).
 Properties :
 a. log_b(b) = 1
 b.log_b(1) = 0
 c.log_b(mn) = log_b(m) + log_b(n)
 d. log_b(^m/_n) = log_b(m) – log_b(n)
 e. log_b(m) = 1/log_m(b)
 f.  log_b(mⁿ) = n · log_b(m)
 g.log_b(m)= log_a(m)/log_a(m)
 Logarithms for base 1 does not exist.Remember that, when base is not mentioned,it is taken as 10.

 In less formal terms, the log rules might be expressed as:

1) Multiplication inside the log can be turned into addition outside the log, and vice versa.
2) Division inside the log can be turned into subtraction outside the log, and vice versa.
3) An exponent on everything inside a log can be moved out front as a multiplier, and vice versa.

Warning: Just as when you're dealing with exponents, the above rules work only if the bases are the same. For instance, the expression "log_d(m) + log_b(n)" cannot be simplified, because the bases (the "d" and the "b") are not the same, just as x² × y³ cannot be simplified (because the bases x and y are not the same).

Expanding logarithms

Log rules can be used to simplify expressions, to "expand" expressions, or to solve for values. 

Expand log₄( ¹⁶/_x ). I have division inside the log, which can be split apart as subtraction outside the log, so

    log₄( ¹⁶/_x ) = log₄(16) – log₄(x)

The first term on the right-hand side of the above equation can be simplified to an exact value, by applying the basic definition of what a logarithm is: 

        log₄(16) = 2

Then the original expression expands fully as: 

      log₄( ¹⁶/_x ) = 2 – log₄(x)

Always remember to take the time to check to see if any of the terms in your expansion (such as the log₄(16) above) can be simplified.

Expand the following:  
The 5 is divided into the 8x⁴, so split the numerator and denominator by using subtraction:
        
Don't take the exponent out front yet; it is only on the x, not the 8, and you can only take the exponent out front if it is "on" everything inside the log. The 8 is multiplied onto the x⁴, so split the factors by using addition:      
       log₂(8x⁴) – log₂(5) = log₂(8) + log₂(x⁴) – log₂(5)
The x has an exponent (which is now "on" everything inside its log), so move the exponent out front as a multiplier: 
        log₂(8) + log₂(x⁴) – log₂(5)  =   log₂(8) + 4log₂(x) – log₂(5)
Since 8 is a power of 2, I can simplify the first log to an exact value:
       log₂(8) + 4log₂(x) – log₂(5) = 3 + 4log₂(x) – log₂(5)
Each log contains only one thing, so this is fully simplified. The answer is: 3 + 4log₂(x) – log₂(5)

some simple problems
A. Solve log₂(8) = x.
I can solve this by converting the logarithmic statement into its equivalent exponential form, using The Relationship: 

         log₂(8) = x  

         2 ^x = 8

  But 8 = 2³, so:2 ^x = 2³  x = 3

Note that this could also have been solved by working directly from the definition of a logarithm: What power, when put on "2", would give you an 8? The power 3, of course!

If you wanted to give yourself a lot of work, you could also do this one in your calculator, using the change-of-base formula:

             log₂(8) = ln(8) / ln(2)

Plug this into your calculator, and you'll get "3" as your answer. While this change-of-base technique is not particularly useful in this case, you can see that it does work. (Try it on your calculator, if you haven't already, so you're sure you know which keys to punch, and in which order.) You will need this technique in later problems.B. Solve log₂(x) + log₂(x – 2) = 3
I can't do anything yet, because I don't yet have "log equals a number". So I'll need to use log rules to combine the two terms on the left-hand side of the equation: 
                   log₂(x) + log₂(x – 2) = 3  
                   log₂((x)(x – 2)) = 3  
                   log₂(x² – 2x) = 3.Then I'll use The Relationship to convert the log form to the corresponding exponential form, and then I'll solve the result: 
                log₂(x² – 2x) = 3   
               2³ = x² – 2x   
               8 = x² – 2x  
              0 = x² – 2x – 8   
              0 = (x – 4)(x + 2)   
               x = 4, –2.But if x = –2, then "log₂(x)", from the original logarithmic equation, will have a negative number for its argument (as will the term "log₂(x – 2)"). Since logs cannot have zero or negative arguments, then the solution to the original equation cannot be x = –2.so,the solution is 4.
Keep in mind that you can check your answers to any "solving" exercise by plugging those answers back into the original equation and checking that the solution "works":
          log₂(x) + log₂(x – 2) = 3  
         log₂(4) + log₂(4 – 2) = 3  
          log₂(4) + log₂(2) = 3
Since the power that turns "2" into "4" is 2 and the power that turns "2" into "2" is "1", then we have:
           log₂(4) + log₂(2) = 3  
          log₂(2²) + log₂(2¹) = 3   
             2 + 1 = 3 ,The solution checks. Copyright © Elizabeth Stapel 2002-2011 All Rights Reserved 
C. Solve log₂(log₂(x))   = 1.
This may look overly-complicated, but it's just another log equation. To solve this, I'll need to apply The Relationship twice: 
      log₂(log₂(x)) = 1
       2¹ = log₂(x)  
       2 = log₂(x) x = 2²   
       x = 4,Then the solution is x = 4.
D. Solve log₂(x²)  = (log₂(x))². First, I'll write out the square on the right-hand side: 
log₂(x²) = (log₂(x))²
log₂(x²) = (log₂(x)) (log₂(x))
Then I'll apply the log rule to move the "squared", from inside the log on the left-hand side of the equation, out in front of that log as a multiplier. Then I'll move that term to the right-hand side:
2log₂(x) = [log₂(x)] [log₂(x)]
0 = [log₂(x)] [log₂(x)]  –  2log₂(x)
This may look bad, but it's nothing more than a factoring exercise at this point. So I'll factor, and then I'll solve the factors by using The Relationship:
0 = [log₂(x)] [log₂(x) – 2]
log₂(x) = 0  or  log₂(x) – 2 = 0
2⁰ = x   or  log₂(x) = 2
1 = x  or  2² = x
1 = x  or  4 = x
The solution is x = 1, 4.

Seating arrangement

The process of making a group of people in a prefixed manner is called seating arrangement.
Sitting arrangement problems are common in every competitive examination.these are very simple and scoring area for everyone.if we made one single mistake,causes big lose in score of exam.for this type of problems,no shortcuts available.we have to read carefully and understand clearly every line given in question.
Here,Some general guidelines to solve seating arrangement problems.
1. Have a glance at the problem.by this step you can notify the type of arrangement and size of the group.
2. Next you have  to determine the useful information and classify into definite information, comparative information and negative information.
3. In Definite information you will get exact position of person.
Eg: raju is sitting at extreme left corner
4. In Comparative information you will get the position of person with comparing to another person.
Eg: A is sitting second to the right of B
5. A Negative information does not give any definite information but it gives an idea to eliminate the possibility.
Eg: A is not an immediate neighbor of B

Basically,seating arrangement problems are two types:
Type-I : Linear arrangement
In this type of arrangement, we have to arrange them in a row or line. In linear arrangement generally there are two ways of arrangements.
1. Single row arrangement
2. Two row arrangement
Type-II : Circular arrangement
In this type of arrangement, we have to arrange them in a circular shape.
Some tips to get notify easily,
Tip-1:
Clockwise direction-left of person/object and Anti-Clockwise direction-right of person/object.
( if facing towards center )

Blood Relation

A person who is related to another by birth rather than by marriage.Blood relations of a group of persons are given in jumbled form. In these tests, the success of a candidate depends upon knowledge of the Blood Relations, some of which are summarized below. The questions which are asked in this section depend upon Relation.

Questions on Blood Relationship are related to our day to day life. We are bound by our kit hand kin through a chain of relationships. The examiner defines the simple relationships by using rather complicated set of definitions and expects from us to comprehend these definitions rather quickly.
In order to solve these problems, analyse the given statements carefully and systematically.
For examples :
(i) My father's only child means I (Myself).
(ii) Ritu's husband's father-in-law's only daughter means - Ritu (Herself).

Pay particular attention to the information given in the question itself without your personal biases and preconceived notions and assumptions coming to the fore. Questions on Blood Relationship can be solved by any of the following methods:
(i) Deduction Method and
(ii) Pictorial Method
While attempting questions on Blood Relationship, first read all the pieces of information as quickly as possible and then point out the two persons between whom relationship is to be established. Finally, try to co-relate the given relationships. While concluding relationship between two persons be careful about the sexes of the persons involved. Majority of the students tend to define or derive relationship without caring for sex of the persons. Is it possible to define relationship between two persons without knowing their sex?
 Consider the following illustration: A is the child of P and Q.
From this statement can we conclude that P is the father of A. No, it is not possible. Without knowing the sex of either P or Q, it is not possible to conclude that P is the father of A. What we can conclude from the above statement is that'P and Q are parents of A. Thus, we see that the knowledge about the sex of persons is necessary to conclude relationship between the two persons.
Some Important Tips:
(i) First of all choose the two persons, between whom relationship is to be established.
(ii) Next, pin-point the intermediate relationship i.e., such relationship through which long drawn   relationship can be established between the required persons.
(iii) Finally, conclude the relationship directly between the two persons as per the requirement of the question.
(iv) From a particular name we cannot ascertain the sex (gender) of that person. The name does not always show the gender beyond reasonable doubt.
For example, we often hear the same name for male and female in the Punjabi community.The names, Harvinder, Sukhwinder, Gurinder etc. are used for both the sexes in the Punjabi community.
There are certain other names which are used for both the sexes all over the country. For example, Suman, Kamal etc.

The list given below is quite helpful in recognising some indirect relationships and to remember easily the relations may be divided into two sides as given below:
I. Relations of Paternal side:
1.Father's father→ Grandfather
2.Father's mother→ Grandmother
3.Father's brother→ Uncle
4.Father's sister→ Aunt
5.Children of uncle→ Cousin
6.Wife of uncle→ Aunt
7.Children of aunt→ Cousin
8.Husband of aunt→ Uncle
II. Relations of Maternal side:
1.Mother's father→ Maternal grandfather
2.Mother's mother→ Maternal grandmother
3.Mother's brother Maternal uncle
4.Mother's sister→ Aunt
5.Children of maternal uncle→ Cousin
6.Wife of maternal uncle→ Maternal aunt
Relations:
1.Grandfather’s son»Father or Uncle
2.Grandmother’s son»Father or Uncle
3.Grandfather’s only son»Father
4.Grandmother’s only son»Father
5.Mother’s or father’s mother»Grandmother
6.Son’s wife»Daughter-in-Law
7.Daughter’s husband»Son-in-Law
8.Husband’s or wife’s sister»Sister-in-Law
9.Brother’s son»Nephew
10.Brother’s daughter»Niece
11.Uncle or aunt’s son or daughter»Cousin
12.Sister’s husband»Brother-in-Law
13.Brother’s wife»Sister-in-Law
14.Grandson’sor grand daughter’s daughter»Great grand daughter


The questions on Blood Relationship are asked in various formats but substantially there is no difference between them. However, consider the pattern of the question we are obliged to discuss them separately.
                                                                       TYPE-I
Ex.l. Pointing to a man, a woman said, "His mother is the only daughter of my mother". How is the woman related to man?
(1) Mother            (2) Grandmother             (3) Sister           (4) Daughter            (5) None of these
Ex.2. Pointing to a man, a woman said, "His mother is the daughter of my mother's only daughter, How is the man related to that woman?
(l)Son                (2) Father                (3) Brother                 (4) Grandson              (5) None of these
                                           ANSWERS WITH EXPLANATION
1. (1) The two persons between whom relationship is to established.MAN <----------->WOMEN
Therefore, the woman in question is the mother of man in the photograph.Only daughter of woman's mother means woman (herself). Only daughter of woman's mother is the mother of the man in the photograph.
                                                    PICTORIAL METHOD
                                                        women's mother
                                                                   ^
                                                                    |
                                                                    |     daughter
                                                                    |
                                                                women
                                                                   ^
                                                                    |
                                                                    |      son
                                                                    |
                                                                  man
                                                                            
2. (4) The two persons between whom relationship has to be established.MAN <----------->WOMEN
My (Woman's) mother's only daughter means the woman herself. Woman's daughter is the mother of
man in question. Therefore, the woman (speaker) is the grandmother of the man. Thus, man is the grandson of the woman (speaker).
                                                   PICTORIAL METHOD
                                                        women's mother
                                                                    |
                                                                    |    only daughter
                                                                    |
                                                                women
                                                                    |
                                                                    |      daughter
                                                                    |
                                                           man's mother
                                                                        
                                                               TYPE-II
Ex.1. If A is B's brother, B is C's sister and C is D's father, D is A's...
(1) Brother        (2) Sister      (3) Nephew    (4) Cannot be determined         (5) None of these
Ex.2. P is brother of Q. R is the sister of Q. How P is related to R?
(1) Brother       (2) Sister        (3) Uncle         (4) Data inadequate              (5) None of these
                                        ANSWERS WITH EXPLANATION
1. (4) A is the brother of B. (A is male). B is sister of C. (B is female). C is father of D. (C is male)
The sex of D is not known. A and C are brothers of B. A is the uncle of D. Since, sex of D is not clear, we cannot determine what is D to A.
2. (1) P is brother of Q. (P is male). R is sister of Q. (R is female). Therefore, P is the brother of R.

SOME QUESTIONS TO PRACTICE:
1. Pointing towards Women, Madhav said "I am the only son of his father's one of the sons". How
Women is related to Madhav ?
(1) Nephew       (2) Uncle              (3) Either father or uncle        (4) Father          (5) None of these
2. Pointing to a man in the photograph, a woman said, "He is the only son of my mother's father".
How is the woman related to the man in the photograph?
(1) Niece                (2) Sister               (3) Mother             (4) Daughter               (5) None of these
3. Pointing to a gentleman, Deepak said, "His only brother is the father of my daughter's father", How is the gentleman related to Deepak?
(1) Father        (2) Grandfather           (3) Uncle          (4) Brother-in-law           (5) None of these
4. Pointing towards Neeru, Asha said, "I am the only daughter of her mother's son". How is Neeru related to Asha?
(l)Aunt '             (2) Cousin                (3) Niece               (4) Mother              (5) None of these
5. If Neha says, "Amruta's father Raj is the only son of my father-in-law, Mahesh"; then how Bindu, who is sister of Amruta, is related to Mahesh?
(1) Daughter       (2) Wife         (3) Daughter-in-law           (4) Niece            (5) None of these
6. Pointing towards a photograph of a girl, Rajan said, "She has no sisters or daughters but her mother is the only daughter of my mother". How is the girl in the photograph related to Rajan's mother?
(1) Sister-in-Law (2) Granddaughter (3) Daughter-in-Law (4) Cannot be determined (5) None

Directions (7-10) : Study the following information carefully and answer the questions given below :
Five persons are sitting around dinning table. - K, L,M, N and O - K is the mother of M, who is the wife of O. N is the brother of K and L is the husband of K.
Ex.7. How is L related to O?
(1) Father      (2) Mother-in-law  (3) Brother-in-law      (4) Father-in-law       (5) Niece
Ex.8. How is K related to O?
(1) Sister      (2) Mother            (3) Mother-in-law            (4) Brother-in-law     (5) Aunt
Ex.9. How is N related to L?
(l)Son           (2) Cousin           (3) Brother              (4) Brother-in-law           (5) Uncle
Ex.10. How is M related to L ?
(l)Aunt         (2) Niece              (3) Daughter              (4) Daughter-in-law    (5) Mother

11. Pointing towards a person in a photograph, Raman said, "She is the only daughter of the mother of my brother's sister". How is the person in photograph related to Raman?'
(1) Daughter                 (2) Sister              (3) Wife             (4) Cousin           (5) None of these
12. Pointing to a man in a photograph, a woman said, "His brother's father is the only son of my grandfather". How is the woman related to the man in the photograph?
(1) Sister      (2) Mother            (3) Grandmother         (4) Aunt                        (5) Daughter