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Logarithmic Function

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

                                                  logarithmic-function

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,∞]

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, ∞].

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

Solution:

        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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