Logarithms Questions

FACTS  AND  FORMULAE  FOR  LOGARITHMS  QUESTIONS

EXPONENTIAL FUNCTION

 For every 

$x\in R, e^x=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...+\frac{x^n}{n!}+...$ 

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 

     ${\log }_a\left(b\right)=x; a\ne 1, a>0, b>0$

e.g, $2^5=32 \iff {\log }_2\left(32\right)=5$

(i) Natural Logarithm :  

${\log }_e\left(N\right)$ 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 , ${\log }_e\left(5\right), {\log }_e\left(\frac{1}{81}\right)$ ... 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 ${\log }_{10}\left(100\right), {\log }_{10}\left(248\right), etc$

PROPERTIES OF LOGARITHM

1. ${\log }_a\left(xy\right)={\log }_a\left(x\right)+{\log }_a\left(y\right)$

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

3. ${\log }_x\left(x\right)=1$

4. ${\log }_a\left(1\right)=0$

5. ${\log }_a\left(x^p\right)=p{\log }_a\left(x\right)$

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

7. ${\log }_a\left(x\right)=\frac{{\log }_b\left(x\right)}{{\log }_b\left(a\right)}=\frac{\log \left(x\right)}{\log \left(a\right)}$

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

(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, $\overline{1}, \overline{2}$ etc.

Example :

$\begin{array}{cc}Number & Characteristic\\ 348.25& 2\\ 9.2193& 0\\ 0.03125& \overline{2}\end{array}$