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Logarithms 10 2 = 100the base 10 raised to the power 2 gives 100 2 is the power which the base 10 must be raised to, to give 100 the power = logarithm.

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Presentation on theme: "Logarithms 10 2 = 100the base 10 raised to the power 2 gives 100 2 is the power which the base 10 must be raised to, to give 100 the power = logarithm."— Presentation transcript:

1 Logarithms 10 2 = 100the base 10 raised to the power 2 gives is the power which the base 10 must be raised to, to give 100 the power = logarithm 2 is the logarithm to the base 10 of 100 Logarithm is the number which we need to raise a base to for a given answer to what power must I raise 2 to give an answer of 64?ans =6 written as log 2 64 = 6

2 to what power must I raise 2 to give an answer of 64?ans =6 written as log 2 64 = 6 to what power must I raise 5 to give an answer of 625?ans =4 written as log = 4 to what power must I raise 9 to give an answer of 3?ans =1212 written as log 9 3 = 1 2 log a n = p a p = n base answer = number inside

3 1.log a m + log a n = log a mn 2.log a m - log a n = log a m n 3.n log a m = log a m n 4.log n m = log a m change of base law log a n N.B. The log of a negative is impossible to find

4 Proofs:Law 1 log a m + log a n = log a m n Let log a m = p & log a n = q a p = ma q = n a p. a q =m. n a p + q = m. n base answer = number inside log a m. n = p + q log a m n = log a m + log a n Law 2 log a m - log a n = log a mnmn Let log a m = p & log a n = q a p = ma q = n a p = a q mnmn a p - q = mnmn = p - q mnmn log a = log a m - log a n mnmn log a

5 ( ) n Law 4 n p = log n m = log a m log a n Let log n m = p take logs of both sides log a n p =log a m Law 3 Let log a m = p a p = m n log a m = log a m n We need m n a p = m a pn = mnmn base answer = number inside log a m n = pn log a m n = (log a m) n log a m n = n log a m m p log a n =log a m p =log a m log a n log n m = log a m log a n

6 e.g.1 log 4 64 = x base answer = number inside 4 x =64 4x4x =4343 x = 3 e.g.2 log 2 x = = x x =32 e.g.3 log 4 (5x + 6) = = 5x = 5x = 5x 10 5 = x 2

7 e.g.4 log 3 (2x - 4) = 1 + log 3 (4x + 8) log 3 (2x – 4) – log 3 (4x + 8)= 1 12x + 24 = 2x x = -28 x = x = -2.8 For an unknown power always take logs of both sides e.g.5 6 n = 3200 log 10 6 n = log n log 10 6 = log n = log log 10 6 n =

8 Calculations using log 10 log = 3 as 10 3 = 1000 If we want log 2 32 = log n m = log a m change of base law log a n log log 10 2 e.g.6 log 2 55 = log 10 x log log 10 2 = log 10 x 5.78 = log 10 x base answer = number inside = x x =


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