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Logarithms
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Introduction
Introduction

Definition
:-
We define log as, ay= x than y = logax. in loga
x. both x and a are positive ie. x > 0 and a > 0 and also a1.

Dumb Question
:- Why a cannot be 1 ?
Ans
:- Suppose a is 1 then let us attempt to y such that y = log1x.
Now according to definition of log.
1y= x.
But no matter what power we raise to 1 the answer will be. 1 only so we will never never be able to find y.
Hence a cannot be 1.

Some important formulae
:- (Formulaes marked with * are important. This is not be printed)
1. logaa = 1.2. logany1 = 0.3. logca = logba.logcb.
Why ?

Let logb
a = x and logcb = y
So, by definition, a = bx..................................... (i)
b = cy...................................... (ii)
Using (i) & (ii) a = cxy

Now taking log on both sides.
logca = xylogba.logcb.

Illustration - 1.

Find value of log
210.log102 ?
Using formula 3 we get.
log210.log102 = log1010
Now using formula 1 we get
log1010 = 1Hence log210.log10
2 = 14. loga(m.n) = logam + logan
Why ?

Let logam = xloga
n = ySo, m = ax
and n = ay
[Using definition]m.n = a
x.ay= ax + y=> loga(mn) = logaax + y

= x + y   = log
am + logan.

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