FACTS  AND  FORMULAE  FOR  LOGARITHMS  QUESTIONS

 

 

EXPONENTIAL FUNCTION

 For every 

or  

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

 

 

LOGARITHM

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

     

e.g, 

(i) Natural Logarithm :   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 , 

(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

 

PROPERTIES OF LOGARITHM

1. 

2. 

3. 

4. 

5. 

6. 

7. 

 

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 , 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 :

(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,  etc.

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 = {\color{Black} \frac{\log 512}{\log 5}}  =  {\color{Black} \frac{\log 2^{9}}{\log (\frac{10}{2})}}  ={\color{Black} \frac{9\log 2}{\log10-\log 2 }} ={\color{Black} \frac{(9\times 0.3010)}{1-0.3010 }} ={\color{Black} \frac{2.709}{0.699 }} ={\color{Black} \frac{2709}{699 }} =3.876

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90 25647
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
{\color{Black} \Rightarrow \log (3^{3})=1.431}
3 log 3 = 1.431
log 3 = 0.477
log 9 = {\color{Black} \log (3^{2})} = 2 log 3 = (2 x 0.477) = 0.954

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35 22443
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:

{\color{Black}\log (2^{56})=(56\times0.30103) } =16.85768.

Its characteristics is 16.

Hence, the number of digits in {\color{Black}2^{56} } is 17.

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46 17183
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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12 2888
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, 

 

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17 2796