Definition Logarithm :If m= a x, where a>0 ,a not equal to 1 , m>0 then x is called the logarithm of m to the base a and is written as x = log a m, where m, a, x are real numbers.
Laws of logarithms:
1. Law of Product: log (A.B) =log A +log B
2. Law of Quotient : log(A/B) =log A - log B
3. Law of Exponent : log m n = n log m
Properties:
1)We have m = a x if and only if x = log a m.
2)Logarithms of negative numbers and zero are not defined.
3) a0 =1 Hence, 0 = log a 1
i.e. logarithm of 1 to any base is always zero.
4) a1 =a Hence , 1 = log a a
i.e. logarithm of any number to the same base is 1.
5) x = log a ax
6) If log a m = log a n then m = n
7)If a>1 and m>n then log a m > log a n and conversely.
Write logarithm form:
1) 5 3 = 125
Solution:
3 = log 5 125 (By definition of logarithm)
2) 8 3 =512
Solution:
3 = log 8 512 (By definition of logarithm)
Express in exponential form:
1) log 2 128 = 7
Solution: 128 = 2 7 (By definition of logarithm)
2) log 9 6561 =4
Solution: 6561 = 2 9 (By definition of logarithm)
Simplify :
1) log 5 + 2 log 4
Solution:
=log 5 + log 42 (By Law of Exponent)
=log 5 + log 16
=log (5 . 16) (By law of Product)
=log 80
2) 2 log 7 - log 14
Solution:
=log 7 2 - log 14 (By Law of Exponent)
=log 49 - log 14
=log (49/14) (By law of Quotient)
=log (7/2)
3)log 3 + log 2 - 2 log 5
Solution:
=log 3 + log 2 - log 52 (By Law of Exponent)
=log 3 + log 2 - log 25
=log (3 . 2) - log 25 (By Law of Product)
=log 6 - log 25
=log (6/25) (By Law of Quotient)
4) 2 log 3 - 1/2 log 16 + log 12
Solution:
=log 3 2 - log 16 1/2 + log 12 (By Law of Exponent)
=log 9 - log4 + log 12
=log (9/4) + log 12 (By Law of Quotient)
=log (9/4 .12) (By Law of Product)
=log (9. 3)
=log 27
Prove that:
1) log 540 = 2 log 2 + 3 log 3 + log 5
Solution:
L.H.S = log (540)
=log (4 X 27 X 5)
=log ( 22 X 33 X 5)
=log 22 + log 33 + log 5 (By Law of Product)
=2 log 2 + 3 log 3 +log 5 (By Law of Exponent)
=R.H.S.
Hence, log 540 = 2 log 2 + 3 log 3 + log 5
OR
R.H.S =2 log 2 + 3 log 3 + log5
=log 2 2 + log 33 + log 5 (By Law of Exponent)
=log 4 + log 27 + log 5
=log (4 X 27 X 5) (By Law of Product)
=log 540
=L.H.S.
Hence, log 540 = 2 log 2 + 3 log 3 + log 5
2) log 360 = 3 log 2 + 2 log 3 + log 5
R.H.S =3 log 2 + 2 log 3 + log5
=log 2 3 + log 32 + log 5 (By Law of Exponent)
=log 8 + log 9 + log 5
=log (8 X 9 X 5) (By Law of Product)
=log 360
=L.H.S.
Hence, log 360 = 3 log 2 + 2 log 3 + log 5
OR
L.H.S = log (360)
=log (8 X 9 X 5)
=log ( 23 X 32 X 5)
=log 23 + log 32 + log 5 (By Law of Product)
=3 log 2 + 2 log 3 + log 5 (By Law of Exponent)
=R.H.S.
Hence, log 360 = 3 log 2 + 2 log 3 + log 5
Evaluate:
1) log 81/log 9
=log 92 / log 9
=2 log 9 / log 9 (By Law of Exponent)
=2
2)log 2 5 / log2 11 - log4 5 / log4 11
Solution :
We know ,Change of Base formula is log a m = log m /log a
=log 5/log 2/log 11/log 2 - log 5/log 4/ log 11 /log 4
=log 5 .
=log 5 / log 11 - log5 / log 11
=0
