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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J.P.Sinha: Teacher of Physics and Maths for IIT-JEE and PMT

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contact me (J.P.Sinha)on my mobile +91 9871 222 426 (it is a New Delhi mobile number).

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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

List of All My Articles:- Here you can find the links to all the articles written by me.