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Basic Properties of Logarithm

Cool goIITian

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24 Jan 2012 21:53:50 IST
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24 Jan 2012 21:53:50 IST
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Basic Properties of Logarithm
Engineering Entrance , JEE Main , BITSAT , JEE Advanced , academic

# Basic Properties of Logarithm

This article is basically for all those students who have just entered in 10th/11th classes. When you start working on some difficult level questions on Mathematics, use of “Logarithm” comes automatically. However, only a basic knowledge of Logarithm is required to gear up initially which I am providing as below :

Definition of Log : logy x = z    means         x = yz

logy x is pronounced as “log of ‘x’ to the base ‘y’ “

Note : a.     ’x’ and ‘y’ MUST be positive.

b.    In most of the cases, “base” (=y) is either “10″ or “e” (Numerically, e=2.718).

1. Basic Properties of Log :
1. logy 1 = 0    (where ‘y’ is any base value)
2. logy y = 1    (where ‘y’ is any value)        =>    log10 10 = 1
3. logy (a * b) = logy a + logy b
4. logy (a / b) = logy a - logy b
5. logy (x)m = m logy x        =>     logy (1/x) = - logy x
6. loge x = 2.303 log10 x
Note : Many a times, loge x is written as ln x (ln = Natural Log)
7. loga x < loga y        where 0 < x < y

Note : The above means that the graph of log is a strictly increasing function.

2. Logarithmic series :

Logarithm of any number can be represented in terms of a Mathematical expression (known as Series).

ln (1 + x) = x – x2/2 + x3/3 – + …… (infinite terms)        [Learn this series]

Note : The above series is a convergent series which means that for a given finite ‘x’, the series will provide a finite value for ln (1 + x).

In cases where “x” (above) is very small compared to “1″, you can also write as :

ln (1 + x) = x

3. Values to be learned :

Even upto 12th class level, following values of log10 are necessary and sufficient :

log10 1 = 0        log10 2 = 0.301        log10 3 = 0.477        log10 4 = 0.602

log10 5 = 0.699        log10 6 = 0.778        log10 7 = 0.845        log10 8 = 0.903

log10 9 = 0.954        loge 2 = 0.693

Try yourself to calculate log10 values for the followings :

log10 15,    log10 1.004,    log10 25,    log10 100,    log10 (1/30),     log10 (Sqrt (2))

New kid on the Block

Joined: 25 Jan 2012 15:58:36 IST
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25 Jan 2012 16:05:26 IST
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1. IT IS VERY EASY IF U LEARN WELL&IT IS CENTRAL SCIENCE

Cool goIITian

Joined: 6 Oct 2011 15:26:25 IST
Posts: 92
27 Jan 2012 15:26:51 IST
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yaa.... that true

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