FACTS  AND  FORMULAE  FOR  LOGARITHMS  QUESTIONS

 

 

EXPONENTIAL FUNCTION

 For every 

xR, ex=1+x+x22!+x33!+...+xnn!+... 

or  ex=n=0xnn!

Here ex is called as exponential function and it is a finite number for every xR.

 

 

LOGARITHM

Let a,b be positive real numbers then ax=b can be written as 

     logab=x;  a1, a>0, b>0

e.g, 25=32 log232=5

 

(i) Natural Logarithm :  

logeN is called Natural logarithm or Naperian Logarithm, denoted by ln N i.e, when the base is 'e' then it is called as Natural logarithm.

e.g , loge5, loge181 ... etc

 

(ii) Common Logarithm :  is called common logarithm or Brigg's Logarithm i.e., when base of log is 10, then it is called as common logarithm.

e.g log10100, log10248, etc

 

PROPERTIES OF LOGARITHM

1. logaxy=logax+logay

 

 2. logaxy=logax-logay

 

3. logxx=1

 

4. loga1=0

 

5. logaxp=plogax

 

6. logax=1logxa

 

7. logax=logbxlogba=logxloga

 

CHARACTERISTICS AND MANTISSA


Characteristic : The integral part of logarithm is known as characteristic.

Mantissa : The decimal part is known as mantissa and is always positive.

E.g, In logax, the integral part of x is called the characteristic and the decimal part of x is called the mantissa.

For example: In log 3274 = 3.5150, the integral part is 3 i.e., characteristic is 3 and the decimal part is .5150 i.e., mantissa is .5150

To find the characteristic of common logarithm log10x:

(a) when the number is greater than 1  i.e., x > 1

In this case, the characteristic is one less than the number of digits in the left of the decimal point in the given number.

 

(b) when the number is less than 1 i.e., 0<x<1

In this case the characteristic is one more than the number of zeros between the decimal point and the first significant digit of the number and is negative.

Instead of -1, -2, etc. we write, 1¯, 2¯ etc.

Example :

Number Characteristic348.2529.219300.031252¯

Q:

The value of 13log10125 - 2log104 + log1032 :

A) 0 B) 1
C) 2 D) 4/5
 
Answer & Explanation Answer: B) 1

Explanation:
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Q:

It is given that log10 2= 0.301 and log10 3 = 0.477. How many digits are there in (108)10?

A) 19 B) 20
C) 21 D) 22
 
Answer & Explanation Answer: C) 21

Explanation:
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Q:

The value of log2log5625 is :

A) 2 B) 5
C) 10 D) 15
 
Answer & Explanation Answer: A) 2

Explanation:
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Q:

If a = bx, b = cy and c = az then the value of xyz is equal to :

A) -1 B) 0
C) 1 D) abc
 
Answer & Explanation Answer: C) 1

Explanation:
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15 3710
Q:

If 5x - 17 = -x + 7, then x = ?

A) 4 B) 5
C) 7 D) 9
 
Answer & Explanation Answer: A) 4

Explanation:

Given equation  5x-17 = -x+7

Add 1x to each side of the equation

5x-17+x = -x+7+x

6x=24

x=4

 

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