1. if n and a are positive real numbers, and a is not equal to 1, then

If a^{x} = n, then
log _{a} n = x

2. log _{a} n is called logarithmic function. The domain of logarithmic function is positive real numbers and the range is all real numbers.

**3. log of 1 to any base is 0**

log _{b} 1 = 0

**4. log of any number to base as itself is 1**,

log_{ a} a = 1

**5. Logarithm of a Product**

log_{ a} pq = log_{ a} p + log_{a} q

** 6. Logarithm of a Fraction**

log _{a} (p/q) = log _{a} p – log _{a} q

7. log _{a} p^{n} = n log _{a} ^{p}

8. a^{(log a p)} = p

**9. Base Change Rule of Logarithms**:

log _{a} n = log _{b} n × log _{a} b

10.
log _{a} b = (log _{n}b)/(log _{n} a)

**11. if the base is 10, then**

log _{a} b = (log b)/(log a)

12. (log _{a} b) × (log _{b} a ) = 1

13. log _{a} b = 1/log _{b} a

14. log (m + n ) is not equal to log m + log n

15. log _{an} p = 1/n (log _{a} p)

16. log _{an} b^{m} = (m/n)log _{a} b

17. log _{10} b is called
**common logarithm** because the base is 10

18. log _{e} b is called **natural logarithm**. It is denoted by ln b.

19. log _{1/a} b = -log _{a} b

20. log _{a} (1/b) = -log _{a} b

21. log _{1/a} (1/b) = log _{a} b