## Logarithms

### Important Formulas:

1.Let   and  be positive real numbers with . The unique real number  such that  is called the Logarithm of  to the base  and is denoted as  .  is not defined if .

2. Let 





3. 

4. 

5. Logarithmic Function  defined by  is a positive real number is called a Logarithmic Function.

6. Logarithmic function is a bijection from 

7.Types of Logarithms  to the base  are called natural logarithms or Napierean logarithms and are written  or  instead of   Here 

Properties of logarithms

8. 

9.   for any positive real number n.

10.   here  are positive real numbers.

11. If is a positive real number and  is a real number then  

12. Rule for change of base: If    and  and  is positive real number then 

13.  this gives 

14. 

15. 

16. 



17. Logarithmic series and exponential series: If  then  

18. Logarithmic series and exponential series: If  then  

19. Logarithmic series and exponential series: If  then  

20.  is irrational and  the value is approximately 

21.   (approx.)

22.  

23.   

24.