# 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  =  $\frac{\mathrm{log}{2}^{9}}{\mathrm{log}\left(10/2\right)}$  =$\frac{9\mathrm{log}2}{\mathrm{log}10-\mathrm{log}2}$ =$\frac{9*0.3010}{1-0.3010}$ =2.709/0.699 =2709/699 =3.876

101 29508
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$\left({3}^{3}\right)$ = 1.431
3 log 3 = 1.431
log 3 = 0.477
log 9 = log(${3}^{2}$)= 2 log 3 = (2 x 0.477) = 0.954

42 27575
Q:

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

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

Explanation:

$\mathrm{log}\left({2}^{56}\right)$ =56*0.30103 =16.85768.

Its characteristics is 16.

Hence, the number of digits in ${2}^{56}$ is 17.

58 21780
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 $\mathrm{log}\left({4}^{3}\right)=1.8061$

--> 3 log 4 = 1.8061

--> log 4 = 0.6020

--> 2 log 4 = 1.2040

$⇒\mathrm{log}\left({4}^{2}\right)=1.2040$

Therefore, log 16 = 1.2040

21 3924
Q:

What is the characteristic of the logarithm of 0.0000134?

 A) 5 B) -5 C) 6 D) -6

Explanation:

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

16 3707
Q:

The value of $\left(\frac{1}{{\mathrm{log}}_{3}60}+\frac{1}{{\mathrm{log}}_{4}60}+\frac{1}{{\mathrm{log}}_{5}60}\right)is$

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

Explanation:

=> ${\mathrm{log}}_{60}\left(3*4*5\right)$

=>     ${\mathrm{log}}_{60}\left(60\right)$

= 1

10 2698
Q:

Find the logarithm of 144 to the base $2\sqrt{3}$ :

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

Explanation:

17 2676
Q:

A fast moving superfast express crosses another pasenger train in 20 seconds. The speed of faster train is 72 km/hr and speeds of slower train is 27 km/h. Also the length of faster ntrain is 100m, then find the length of the slower train if they are moving in the same direction.

 A) 100 m B) 125 m C) 150 m D) 175 m

$20=\frac{\left(100+x\right)}{25}{2}}$
$⇒$ X=150m