# Logarithms Questions

FACTS  AND  FORMULAE  FOR  LOGARITHMS  QUESTIONS

EXPONENTIAL FUNCTION

For every

or  ${e}^{x}=\sum _{n=0}^{\infty }\frac{{x}^{n}}{n!}$

Here ${e}^{x}$ is called as exponential function and it is a finite number for every $x\in R$.

LOGARITHM

Let a,b be positive real numbers then ${a}^{x}=b$ can be written as

e.g,

(i) Natural Logarithm :

${\mathrm{log}}_{e}\left(N\right)$ 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 ,  ... etc

(ii) Common Logarithm :  $\inline \fn_cm \log_{10}N$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. ${\mathrm{log}}_{a}\left(xy\right)={\mathrm{log}}_{a}\left(x\right)+{\mathrm{log}}_{a}\left(y\right)$

2. ${\mathrm{log}}_{a}\left(\frac{x}{y}\right)={\mathrm{log}}_{a}\left(x\right)-{\mathrm{log}}_{a}\left(y\right)$

3. ${\mathrm{log}}_{x}\left(x\right)=1$

4. ${\mathrm{log}}_{a}\left(1\right)=0$

5. ${\mathrm{log}}_{a}\left({x}^{p}\right)=p{\mathrm{log}}_{a}\left(x\right)$

6. ${\mathrm{log}}_{a}\left(x\right)=\frac{1}{{\mathrm{log}}_{x}\left(a\right)}$

7. ${\mathrm{log}}_{a}\left(x\right)=\frac{{\mathrm{log}}_{b}\left(x\right)}{{\mathrm{log}}_{b}\left(a\right)}=\frac{\mathrm{log}\left(x\right)}{\mathrm{log}\left(a\right)}$

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 ${\mathrm{log}}_{a}\left(x\right)$, 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 ${\mathrm{log}}_{10}\left(x\right)$:

(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.

Example :

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

Explanation:

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

Filed Under: Logarithms
Exam Prep: Bank Exams
Job Role: Bank PO

205 71367
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

Explanation:

log 27 = 1.431
log$33$ = 1.431
3 log 3 = 1.431
log 3 = 0.477
log 9 = log($32$)= 2 log 3 = (2 x 0.477) = 0.954

Filed Under: Logarithms
Exam Prep: Bank Exams
Job Role: Bank PO

127 62799
Q:

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

 A) 17 B) 19 C) 23 D) 25

Explanation:

$log(256)$ =56*0.30103 =16.85768.

Its characteristics is 16.

Hence, the number of digits in $256$ is 17.

Filed Under: Logarithms
Exam Prep: Bank Exams
Job Role: Bank PO

119 57672
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

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

Filed Under: Logarithms
Exam Prep: CAT
Job Role: Bank Clerk

49 16404
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

Explanation:

= $12+122log72$
= $12+log72$

$1+2m2$.

Filed Under: Logarithms
Exam Prep: GRE , GATE , CAT , Bank Exams , AIEEE
Job Role: Bank PO , Bank Clerk , Analyst

61 11681
Q:

Find the logarithm of 144 to the base $23$ :

 A) 2 B) 4 C) 8 D) None of these

Explanation:

Filed Under: Logarithms
Exam Prep: AIEEE , Bank Exams , CAT
Job Role: Analyst , Bank Clerk , Bank PO

55 10843
Q:

What is the number of digits in $333$? Given that log3 = 0.47712?

 A) 12 B) 13 C) 14 D) 15

Explanation:

Let   $333$

Then,

= 27 x 0.47712 = 12.88224

Since the characteristic in the resultant value of log x is 12

$∴$The number of digits in x is (12 + 1) = 13

Hence the required number of digits in $333$is 13.

Filed Under: Logarithms
Exam Prep: AIEEE , Bank Exams , CAT
Job Role: Bank Clerk , Bank PO

49 10241
Q:

The value of $1log360+1log460+1log560is$

 A) 0 B) 1 C) 5 D) 60

Explanation:

=> $log60(3*4*5)$

=>     $log6060$

= 1