Logarithms - Basic points

Let a,b be two positive real numbers and a≠1. The real number x such that a^x = b is called logarithm of b to the base a.

X = log_ab

Theorems

1. If a is a positive number and a≠1 then log_aa = 1.

2. If a is a positive number and a≠1 then log_a1 = 0

3. If a,m are positive real numbers, a≠1 then a to the power log_am = m.

4. If a,m,n are positive real numbers and a≠1 then log_amn = log_am + log_an

5. If a,m,n are positive real numbers and a≠1 then log_am/n = log_am - log_an

6. If a,m,n are positive real numbers and a≠1 then log_amⁿ = nlog_am

7. If a,b,n are positive real numbers and a≠1, b≠1 then log_am = log_bm* log_ab

8. If a>1, then x>y => log_ax > log_ay

9. If 0y => log_ax < log_ay




















e = 1+1/1! +1/2! + 1/3! + ¼! + …∞

e = lim n →∞ (1+(1/n)) ⁿ


If aⁿ = x, the log_ax = n.

Log (1=x) when |x|<1 = x-x²/2 +x³/3 - x⁴/4+…∞