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Logarithms 102 = 100 the 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 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 log2 64 = 6 to what power must I raise 2 to give an answer of 64? ans = 6 written as log2 64 = 6 to what power must I raise 5 to give an answer of 625? ans = 4 written as log5 625 = 4 to what power must I raise 9 to give an answer of 3? written as log9 3 = 1 2 loga n = p ans = 1 2 ap = n baseanswer = number inside 1. loga m + loga n = loga mn 2. loga m - loga n = loga m n 3. n loga m = loga mn 4. logn m = loga m loga n change of base law N.B. The log of a negative is impossible to find Proofs: Law 1 loga m + loga n = loga m n Let loga m = p & loga n = q ap = m aq = n ap . aq = m . n ap + q = m.n loga m . n = p + q loga m n = loga m + loga n baseanswer = number inside Law 2 loga m - loga n = loga m n Let loga m = p & loga n = q ap = m ap aq aq = n = m n ap - q = m n loga m = p - q n loga m = loga m - loga n n Law 3 n loga m = loga mn Let loga m = p ap = m p )n (a apn loga mn We need mn = ( m )n = mn = pn baseanswer = number inside loga mn = (loga m) n loga mn = n loga m Law 4 logn m = loga m loga n Let logn m = p np = m take logs of both sides loga np = loga m p loga n = loga m p = loga m loga n logn m = loga m loga n e.g.1 log4 64 = x x 4 = 64 4 x = 43 x = 3 e.g.2 log2 x = 5 25 = x x = 32 e.g.3 log4 (5x + 6) = 2 baseanswer = number inside 42 = 16 = 10 = 10 = 5 5x + 6 5x + 6 5x x 2 = x e.g.4 log3 (2x - 4) = 1 + log3 (4x + 8) log3 (2x – 4) – log3 (4x + 8) = 1 2x 4 log3 1 4x 8 2x 4 3 4x 8 1 3(4 x 8) 2 x 4 12x + 24 = 2x - 4 10x = -28 x = -28 10 x = -2.8 For an unknown power always take logs of both sides e.g.5 6n = 3200 log10 6n = log10 3200 n log10 6 = log10 3200 n = log10 3200 log10 6 n = 4.5045 Calculations using log10 log10 1000 = 3 as 103 = 1000 If we want log2 32 = log10 32 logn m = loga m loga n change of base law log10 2 e.g.6 log2 55 = log10 x log10 55 = log10 x log10 2 5.78 = log10 x 105.78 = x x = 604449 baseanswer = number inside