Logarithm and exponential functions

Properties of Logarithmic Function:

log (ab) = log a + log b         (a, b > 0)

log(a/b) = log a – log b        (a, b > 0)

log a^m = m log a                     (a  > 0, m ∊ R)

log_a a = 1                                  (a > 0, a ≠ 1)

                 (a,b > 0, b ≠ 1, m & n ∊ R – {0})

log_b a = (1/log_a b)                  (a, b > 0, a ≠ 1)

log_b a = (log_ma)/(log_mb)      (a, b, m > 0, a ≠ 1)

                        (a > 0, a ≠ 1; m > 0)

                        (a, b, c > 0, c ≠ 1)

log_a 1 = 0                                  (a > 0, a ≠ 1)

Properties of Exponential Function:

a^x = a^y ⇔ x = y

a^x = b^x ⇔ a = b

a^x.a^y = a^x+y

a^x.b^x = (a.b)^x (b > 0)

(a^x)^y = a^xy

(a^x/a^y) = a^x–y

Exponential Equation:

► a^f(x) = b (a > 0)

(a) x ∊ φ if b ≤ 0

(b) f(x) = log_a b if b > 0, a ≠ 1

(c) Equation is satisfied ∀ x ∊ Domain of ƒ if a = b = 1

Exponential Inequalities:

Logarithmic Inequalities:

(a)

(b)

(c)