Dictionary Rank of a Word | Permutations & Combinations

 PERMUTATIONS & COMBINATIONS Rank of the word or Dictionary order of the English words like COMPUTER, COLLEGE, SUCCESS, SOCCER, RAIN, FATHER, etc. Dictionary Rank of a Word Method of finding the Rank (Dictionary Order) of the word  “R A I N” Given word: R A I N Total letters = 4 Letters in alphabetical order: A, I, N, R No. of words formed starting with A = 3! = 6 No. of words formed starting with I = 3! = 6 No. of words formed starting with N = 3! = 6 After N there is R which is required R ----- Required A ---- Required I ---- Required N ---- Required RAIN ----- 1 word   RANK OF THE WORD “R A I N” A….. = 3! = 6 I……. = 3! = 6 N….. = 3! = 6 R…A…I…N = 1 word 6 6 6 1 TOTAL 19 Rank of “R A I N” is 19 Method of finding the Rank (Dictionary Order) of the word  “F A T H E R” Given word is :  "F A T H E R" In alphabetical order: A, E, F, H, R, T Words beginni

Introduction to logarithms


Introduction to logarithms

Simple explanations and the implementations of logarithmic formulas in solving complicated mathematical problems. Indices, logarithmic formulas and logarithmic tables are explained here

Logarithmic Tables
******************

Indices

xm = x.x.x.x…………….m terms

Here x is called the base and m is called the index or exponent of x.

Indices is the plural of index.

\[If\;\; \; \; x^{2}=a\; \; \; \Rightarrow\; \; \; x=\sqrt{a}\]

\[If\;\; \; \; x^{3}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[3]{a}\]

\[If\;\; \; \; x^{4}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[4]{a}\]

\[If\;\; \; \; x^{5}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[5]{a}\]

\[If\;\; \; \; x^{100}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[100]{a}\]

\[If\;\; \; \; x^{n}=a\; \; \; \Rightarrow\; \; \; x= \sqrt[n]{a}\]

Laws of Indices

\[(i)\; \; x^{m}\times x^{n}=x^{m+n}\; \; \; (ii)\; \; \; \; \frac{x^{m}}{x^{n}}=x^{m-n}\]

\[(iii)\; \; (x^{m})^{n}=x^{mn}\; \; \; (iv)\; \; \; \; x^{m}=\frac{1}{x^{-m}}\; \; or\; \; x^{-m}=\frac{1}{x^{m}}\]

\[(v)\; \; \; \; (xy)^{m}=x^{m}y^{m}\; \;\: \; (vi)\; \; \; \left ( \frac{x}{y} \right )^{m}=\frac{x^{m}}{x^{n}}\]

\[(vii)\; \; \; \; x^{a}=y^{a}\; \;\Rightarrow \: \;x=y,\; \;Provided\; \; a\neq 0\]

\[(vii)\; \; \; \; x^{m}=x^{n}\; \;\Rightarrow \: \;m=n,\; \;Provided\; \; x\neq \: 0,1\]

Logarithms

Logarithm of a number to a given base is the index or power to which the base must be raised so as to be equal to the given number.

If  a x = N (Where a is a positive number ≠1), then x is called the logarithm of N to the base a and is written as  x = Loga N.

\[For\; Example:\; \; 5^{3}=125, \Rightarrow 3=Log_{5}125\]

\[ 4^{5}=1024, \Rightarrow 5=Log_{4}1024\]

\[Log1=0,\; \; log10=1, \; \; log100=2,\; \; log1000=3\; \; ....so \; \; on.\]

\[Log_{a}1=\frac{Log1}{Loga}=\frac{0}{Loga}=0\]

\[Log_{a}a=\frac{Loga}{Loga}=1\]


Example: \[If\; \; x = 2^{-\frac{1}{3}log_{2}64},\; \; then\; find\; \; x\]

Solution: \[x = 2^{-\frac{1}{3}log_{2}64}\; \; =2^{log_{2}\left ( 64 \right )^{-1/3}}\]

\[=2^{log_{2}\left ( 4^{3} \right )^{-1/3}}=2^{log_{2}(4)^{-1}}\]

\[=2^{log_{2}(2^{2})^{-1}}=2^{log_{2}(2)^{-2}}=2^{-2log_{2}2}\]

\[=2^{-2\times 1}=2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}\]


Fundamental Properties of Logarithms

Product Formula:  \[Log_{a}(lm)=Log_{a}l+Log_{a}m\]

\[Log_{a}(lmn)=Log_{a}\: l+Log_{a}\: m+Log_{a}\: n\]

Quotient Formula:  \[Log_{a}(\frac{m}{n})=Log_{a}\: m-Log_{a}\: n\]

Power formula: \[Log_{a}m^{n}=n\: Log_{a}\: m\]


Example: \[Evaluate:\: \: Log_{5}\; \left ( \frac{\sqrt[4]{25}}{625} \right )\]

Solution:  \[Log_{5}\; \left ( \frac{\sqrt[4]{25}}{625} \right )=Log_{5}\sqrt[4]{25}-Log_{5}\; 625\]

\[=Log_{5}\; (25)^{1/4}-Log_{5}\; 5^{4}=\frac{1}{4}Log_{5}\; (25)-4Log_{5}5\]

\[=\frac{1}{4}Log_{5}\; (5)^{2}-4\times1 =\frac{1}{4}\times 2\; Log_{5}\; (5)-4\]

\[=\frac{1}{2}\times 1-4=\frac{-7}{2}\]

Natural Logarithms

Logarithms to the base e (= 2.7183) are called natural logarithms. They are used in all theoretical calculations. For Example :  Log e (sin x)  This logarithm is called the natural logarithm.

Common Logarithms
Logarithms to the base 10 are called common logarithms. They are used in arithmetic calculations.  For Example :   Log 10 (34.56)  This logarithm is called the common logarithm.

Characteristic and Mantissa

The integral part of the logarithm of a number, after expressing the decimal part as positive (if not already so), is called the characteristic.
Mantissa
Positive decimal part is called Mantissa. Note: Mantissa is always positive.

Rule to find the Characteristic
Rule 1
  • Characteristic of the logarithm of a number > 1 is positive
  • Characteristic of the logarithm of a number is one less than the number of digits in the integral part of the number.
For Example :  34.586
  • Integral Part Before Decimal = 34
  • No. Digits in the integral part = 2
  • Characteristic = 2 - 1 = 1
Rule II
  • Characteristic of the logarithm of a number < 1 is negative.
  • It is numerically one more than the number of zeroes immediately after the decimal point.
For Example :  0.0586  
Here given number is  < 1 so characteristic should be negative
Number of zeroes after decimal = 1
Characteristic = - (1 + 1) = - 2
Negative character always written in bar form as given below  \[-2 \: \: is\: written\: in\: the\: form\: of\: bar\: as\: \: \overline{2}\]

Characteristic and Mantissa for a negative number
For Example:  
Let the number is   - 2.305
- 2.305 = - (2 + 0. 305) = - 2 - 0.305 
To make decimal part positive we have to subtract and add 1 to this number
- 2 - 1 + 1 - 0.305 = - 3 + 0.695 \[=\overline{3}.695\]

Find the number of digits in  362

Let  x =   362

Taking log on both side

Log x = Log 362

Log x = 62 Log 3 = 62 X 0.4771 = 29.5802

Characteristic  = 29

Number of digits in  362 = 29 + 1 = 30

Find the position of the first significant figure in  3-65

Let x = 3-65 

Log x = Log 3-65  = - 65 X Log 3  = -65 X 0.4771 = -31.0115

Log x = -(31 + 0.0115) = -31 – 0.0115

In order to make Mantissa positive ,  add and subtract 1

Log x =  - 31 – 1 + 1 -0.0115

           =  - 32 + 0.9985

Characteristic Value = - 32

So there are 32 – 1= 31 zeroes immediate after decimal.

So position of the first significant figure is  31 + 1 = 32

Find the 5th root of 0.003

\[Let\: \: x=\left ( 0.003 \right )^{1/5}\]

\[Logx=Log\left ( 0.003 \right )^{1/5}\]

\[Log x=\frac{1}{5}\; Log \left ( 0.003 \right )\]

\[Log x=\frac{1}{5}\times \overline{3}.4771\: =\: \frac{1}{5}[-3+0.4771]\]

\[Log x=\: \frac{1}{5}[-3-2+2+0.4771]=\: \frac{1}{5}[-5+2.4771]\]

\[Log x=\: \frac{1}{5}\times -5+\frac{1}{5}\times 2.4771 = -1+0.4954 = \overline {1} .4954\]

Taking antilog on both sides we get

\[x=Antilog(\overline{1}.4954)=0.3129\]

Find the value of  Log 43 57

\[Let \; \; x = Log_{43}57 = \frac{Log57}{Log43}=\frac{1.7559}{1.6335}\]

\[Log x = Log\left (\frac{1.7559}{1.6335} \right )=Log1.7559-Log1.6335\]

\[Log x = 0.2443-0.2130=0.0312\]

Taking antilog on both sides we get

\[x = Antilog\: (0.0312)=1.074\]

Evaluate the following\[Evaluate\: :\: \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\] Solution : \[Let\; \; x=\: \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]

Taking log on both side we get
\[Log\: x=Log \sqrt[3]{\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}}}\]
\[Log\: x=Log \left (\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}} \right )^{1/3}\]
\[Log\: x=\frac{1}{3}Log \left (\frac{(45.4)^{2}}{(3.2)^{2}\times (6.5)^{3}} \right )\]
\[Log\: x=\frac{1}{3}\left [Log(45.4)^{2}-log(3.2)^{2}-log(6.5)^{3} \right ]\]
\[Log\: x=\frac{1}{3}\left [2Log(45.4)-2log(3.2)-3log(6.5) \right ]\]
\[Log\: x=\frac{1}{3}\left [2\times 1.6571-2\times0.5051-3\times 0.8129 \right ]\]
\[Log\: x=\frac{1}{3}\left [3.3142-1.0102-2.4387 \right ]\]
\[Log\: x=\frac{1}{3}\left [3.3142-3.4489 \right ]\]
\[Log\: x=\frac{1}{3}\times \left (-0.1347 \right )\]
\[Log\: x=\frac{1}{3}\times \left (-3+3-0.1347 \right )=\frac{1}{3}\times \left (-3+2.8653 \right )\]
\[Log\: x=-1+0.9551=\overline{1}.9551\]
Taking Antilog on both side we get
\[x=Antilog\left (\overline{1}.9551 \right )\]
\[x=Antilog\left (\overline{1}.9551 \right ) = 0.9018\; Ans\]

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