# Logarithms Questions

FACTS  AND  FORMULAE  FOR  LOGARITHMS  QUESTIONS

EXPONENTIAL FUNCTION

For every $\inline \fn_cm x\epsilon R, e^{x}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}+...\frac{x^{n}}{n!}+...$

or  $\inline \fn_cm e^{x}=\sum_{n=0}^{\infty }\frac{x^{n}}{n!}$

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

LOGARITHM

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

$\inline \fn_jvn \log_{a}b=x;\; \; a\neq 1, a>0,b>0$

e.g, $\inline \fn_cm 2^{5}=32\Leftrightarrow \log_2{32}=5$

(i) Natural Logarithm :  $\inline \fn_cm \log_e{N}$ 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 , $\inline \fn_cm \log_e{5}, \log_e{\frac{1}{81}}...etc$

(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

$\inline \fn_cm \log_{10}100, \log_{10}248 \; etc$

PROPERTIES OF LOGARITHM

1. $\inline \fn_cm \log_{a}(xy)=\log_{a}x+\log_{a}y$

2. $\inline \fn_cm \log_{a}(\frac{x}{y})=\log_{a}x-\log_{a}y$

3. $\inline \fn_cm \log_{x}x=1$

4. $\inline \fn_cm \log_{a}1=0$

5. $\inline \fn_cm \log_{a}(x^{p})=P(\log_{a}x)$

6. $\inline \fn_cm \log_{a}(x)=\frac{1}{\log_{x}a}$

7. $\inline \fn_cm \log_{a}(x)=\frac{\log_{b}x}{\log_{b}a}=\frac{\log x}{\log a}$

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 $\inline \fn_cm \log_{a}x$, 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 $\inline \fn_cm \log_{10}x$:

(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, $\inline \fn_cm \bar{1}, \bar{2}$ etc.

$\inline \fn_cm \begin{matrix} Number & Characteristic\\ 348.25&2 \\ 9.2193&0 \\ 0.6173& \bar{1}\\ 0.00125& \bar{3} \end{matrix}$

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

87 24720
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
${\color{Black}&space;\Rightarrow&space;\log&space;(3^{3})=1.431}$
3 log 3 = 1.431
log 3 = 0.477
log 9 = ${\color{Black}&space;\log&space;(3^{2})}$ = 2 log 3 = (2 x 0.477) = 0.954

34 21693
Q:

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

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

Explanation:

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

Its characteristics is 16.

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

44 16295
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.

11 2749
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, $\inline \fn_jvn \small \log 64=1.8061$

$\inline \fn_jvn \small i.e. \log 4^{3}=1.8061$

$\inline \fn_jvn \small \Rightarrow 3\log 4=1.8061$

$\inline \fn_jvn \small \Rightarrow \log 4=0.6020$

$\inline \fn_jvn \small \Rightarrow 2\log 4=1.2040$

$\inline \fn_jvn \small \Rightarrow \log 4^{2}=1.2040$

$\inline \fn_jvn \small \Rightarrow \log 16=1.2040(approx)$