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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, 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. AskIITians offers a novel way of teaching where you can prepare for IIT JEE, AIEEE and other engineering examinations for free by sitting at home. You can visit the website askIITians.com to read the study material pertaining to your preparation. Be a part of our online tests and AQAD (A Question A Day) for free and be a winner. To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
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.
AskIITians offers a novel way of teaching where you can prepare for IIT JEE, AIEEE and other engineering examinations for free by sitting at home. You can visit the website askIITians.com to read the study material pertaining to your preparation. Be a part of our online tests and AQAD (A Question A Day) for free and be a winner.
To read more, Buy study materials of Set Relations and Functions comprising study notes, revision notes, video lectures, previous year solved questions etc. Also browse for more study materials on Mathematics here.
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