Properties of Logarithm

Below are some properties of Logarithmic algebra of positive numbers.

(i) Product rule: If a, m and n are positive numbers and a ≠1, then

log_a (mn) = log_a m + log_a n

(ii) Quotient rule: If m, n and a are positive numbers and a ≠ 1, then,

log _a(m/n) = log_a m - log_a n

(iii) Power rule: If a and m are positive numbers, a ≠ 1 and n is a real number, then

log_a mⁿ = n log_a m

(iv) Change of base rule: If m, n and p are positive numbers and n ≠ 1, p ≠ 1, then

log_n m = (log _p m) (log_np)

(v) Reciprocal rule: If m and n are positive numbers other than 1, then

log_nm = 1/(log_mn)

(vi) If a is a positive number, then log_a 1 = 0

(vii) If a is a positive number, then log _aa = 1

(viii) If a and b are any two positive numbers and b ≠ 1, then b log_b a = a

(ix) Let m, n and a be positive numbers and a ≠ 1.log am = log an = then m = n.