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We have observed that y = ax is a monotonic function (either strictly) decreasing or strictly increasing). Hence it is invertible,
So y = ax <=> x = loga y
Where x ε [-∞,∞ ] and y ε [0, ∞]
The inverse exponential function x = loga y is known as logarithmic function. Writing it in conventional form it becomes
y = loga x = f(x), x ε [0,∞]
y = loga x = f(x), x ε [0, ∞].
Properties of logarithmic Function:
(i) y = logb x is defined for x > 0, b > 0, b ≠ 1.
(ii) if logb a = c then a = bc
(iii) logb 1 = 0
(iv) logb b = 1
(v) logb a = 1/loga b
(vi) logb xy = logb x + logb y
(vii) logb XY = logb x - logb y
(viii) logb xm = m logb x
(ix) logbn x = 1/n logb x
(x) logb bx = x
(xi) (b)logbx = x
Illustration: Prove logb a = 1/loga b
Solution:
Let c = logb a and d = loga b
=> a = bc and b = ad
=> a = bc and a = b(1/d)
=> c = 1/d
=> logb a = 1/loga b
Illustration: Prove logb xm = m logb x
Let c = logb xm and d = logb x
=> xm = bc and x = (b)d
=> ((b)d)m = bc
=> md = c
=> logb xm = m logb x.
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