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:

If log 27 = 1.431, then the value of log 9 is

A) 0.754 B) 0.854
C) 0.954 D) 0.654
 
Answer & Explanation Answer: C) 0.954

Explanation:

log 27 = 1.431
log33 = 1.431
3 log 3 = 1.431
log 3 = 0.477
log 9 = log(32)= 2 log 3 = (2 x 0.477) = 0.954

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48 34395
Q:

If log 2 = 0.3010 and log 3 = 0.4771, the values of log5 512 is

A) 2.875 B) 3.875
C) 4.875 D) 5.875
 
Answer & Explanation Answer: B) 3.875

Explanation:

ANS:      log5512 = log512/log5  =  log29log10/2  =9log2log10-log2 =9*0.30101-0.3010 =2.709/0.699 =2709/699 =3.876

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114 33608
Q:

 If log 2 = 0.30103, Find the number of digits in 256 is

A) 17 B) 19
C) 23 D) 25
 
Answer & Explanation Answer: A) 17

Explanation:

log(256) =56*0.30103 =16.85768.

 

Its characteristics is 16.

 

Hence, the number of digits in 256 is 17.

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71 27083
Q:

If log 64 = 1.8061, then the value of log 16 will be (approx)?

A) 1.9048 B) 1.2040
C) 0.9840 D) 1.4521
 
Answer & Explanation Answer: B) 1.2040

Explanation:

Given that, log 64 = 1.8061

 i.e log43=1.8061 

--> 3 log 4 = 1.8061

--> log 4 = 0.6020

--> 2 log 4 = 1.2040

log42=1.2040

Therefore, log 16 = 1.2040

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24 5354
Q:

What is the characteristic of the logarithm of 0.0000134?

A) 5 B) -5
C) 6 D) -6
 
Answer & Explanation Answer: B) -5

Explanation:

log (0.0000134). Since there are four zeros between the decimal point and
the first significant digit, the characteristic is –5.

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17 4707
Q:

Find the logarithm of 144 to the base 23 :

A) 2 B) 4
C) 8 D) None of these
 
Answer & Explanation Answer: B) 4

Explanation:

log23144 = 4 

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19 3573
Q:

The value of 1log360+1log460+1log560is

A) 0 B) 1
C) 5 D) 60
 
Answer & Explanation Answer: B) 1

Explanation:

            => log60(3*4*5)

                              =>     log6060

                                   = 1

 

                                                       

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10 3512
Q:

If log72 = m, then log4928 is equal to ?

A) 1/(1+2m) B) (1+2m)/2
C) 2m/(2m+1) D) (2m+1)/2m
 
Answer & Explanation Answer: B) (1+2m)/2

Explanation:

log4928 = 12log77×4

 

= 12+122log72
= 12+log72
12 + m
1+2m2.

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26 3061