What is a Perfect Number ?

Perfect Number

Let us consider the number 6, what are the numbers (positive integers) that can fully divide 6 without leaving any remainder ? Those numbers are : 1 , 2, 3 and 6. These are called factors of 6. Obviously 1 is a factor of every number, and every number is a factor of itself. Among the factors of 6 the numbers 1,2, and 3 are called Proper Factors of 6. So when we say proper factor of 6, we mean a factor of 6 other than 6 itself.

Now if we take all the proper factors of 6 and add them, what do we get ?

1+2+3=6

Thus we see that the sum of all the proper factors of 6 is equal to 6.

Also consider the number 28. Let us make a list of all its divisors ( factors). These are : 1,2,4,7,14 and 28.

And we see 1+2+4+7+14=28

Thus the sum of all the proper factors of 28 is 28

Numbers like 6 and 28 are called Perfect Numbers. A number that is equal to the sum of all its proper factors is called Perfect Number.

Next two perfect numbers are 496 and 8128.

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Integrating ROOT of tanx

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Logarithm-3

USEFUL FORMULAE OF LOGARITHM

1. log_a(mn)=log_am + log_an
2. log_a(m/n) = log_am - log_an
3. log_amⁿ = nlog_am
4. log_bm = log_am/log_ab
5. (log_ab)(log_ba) = 1
  

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LOGARITHM-2

In my last article (Logarithm-1) I explained to you that what does it mean to take the logarithm of a number x to some base a. I explained there that when we take the logarithm of x to base a, we try to find out that to what power we must raise a to get x. In this sense if we have decided the value of a then for every value of x we will get a value y such that a^y =x. And we write log_a(x)=y.

For example let us take a=10.

Then following table illustrates the changing values of y with the changing values of x.

X	y= log10(x)	Explanation
0.01	-2	10-2=0.01
0.1	-1	10-1=0.1
1	0	100=1
10	1	101=10
100	2	102=100
1000	3	103=1000

So we see that for every value of x , log₁₀(x) gives an unique value of y. This kind of arrangement in mathematics is known as FUNCTION. Thus we can treat LOGARITHM as a function.

Therefore, now onwards we will treat y= log_a(x) as a FUNCTION.

Rules for the values of base a, argument x and dependent variable y:

a ε (0,+∞) - {1}. Base a can take all positive values except 1.

x ε (0, +∞) ; Domain of logarithmic function.

y ε (- ∞,+∞) ; Range of logarithmic function.

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LOGARITHM-1

LOGARITHM

Author: J.P.Sinha, a teacher of Physics, Maths for IIT-JEE/PMT


These days, logarithm has strangely been dropped from the school syllabus of CBSE-system. Thus depriving the students from learning a very important skill in mathematics. Logarithm is used to expedite the solution of complicated arithmetical calculations. It is also an important FUNCTION that gives useful formulae in Integration and Differentiation.

DEFINITION:

We know 2⁵=32 , so if we are asked 2^?=32 then obviously our answer will be 5.
When be write 2^?=32, we mean "to what power 2 must be raised to get 32?".
In mathematics normally we ask this question by using a different wording. We ask :
What is the logarithm of 32 to base 2?.And in symbols we express this question
as log₂ 32=?

Thus,

"to what power 2 be raised to give 32? ( i.e. 2^?=32)"
and
" what is the logarithm of 32 to base 2? (i.e. log₂ 32=?)"

are two different ways of asking the same thing.

So, whenever we write " log_a b =? " we mean "a^?=b".

And if, log_a b =c , then that will mean a^c=b.

The expression log_a b =c is read as " the logarithm of b to base a is c.

Now watch the following video that I have recorded to explain the basic idea Logarithm

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