We have observed that y = a^x is a monotonic function (either strictly) decreasing or strictly increasing). Hence it is invertible,

So y = a^x <=> x = log_a y

Where x ε [-∞,∞ ] and y ε [0, ∞]

        The inverse exponential function x = log_a y is known as logarithmic function. Writing it in conventional form it becomes

y = log_a x = f(x), x ε [0,∞]

        The inverse exponential function x = log_a y is known as logarithmic function. Writing it in conventional form it becomes

y = log_a x = f(x), x ε [0, ∞].

Properties of logarithmic Function:

(i)     y = log_b x is defined for x > 0, b > 0, b ≠ 1.

(ii)    if log_b a = c then a = b^c

(iii)    log_b 1 = 0

(iv)   log_b b = 1

(v)    log_b a = 1/log_a b

(vi)   log_b xy = log_b x + log_b y

(vii)   log_b XY = log_b x - log_b y

(viii)  log_b x^m = m log_b x

(ix)   log_bn x = 1/n log_b x

(x)    log_b b^x = x

(xi)   (b)^logbx = x

Illustration: Prove log_b a = 1/log_a b

Solution:

        Let c = log_b a      and   d = log_a b

        =>     a = b_c        and   b = a_d

        =>     a = b_c        and   a = b^(1/d)

        =>     c = 1/d

        =>     log_b a = 1/log_a b

Illustration: Prove log_b x^m = m log_b x

Solution:

        Let    c = log_b x^m         and   d = log_b x

        =>     x^m = b^c              and   x = (b)^d

        =>     ((b)^d)^m = b^c

        =>     md = c

        =>     log_b x^m = m log_b x.