### 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
$x=Antilog\left (\overline{1}.9551 \right )$
$x=Antilog\left (\overline{1}.9551 \right ) = 0.9018\; Ans$