Logarithms 102 100 the base 10 raised to

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 log 2 64 = 6

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 5 625 = 4 to what power must I raise 9 to give an answer of 3? written as log 9 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. a q = 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 We need mn ( ap ) n = ( m )n apn mn loga 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 log 4 64 = x x 4 = 64 4 x = 43 x = 3 e. g. 2 log 2 x = 5 25 = x x = 32 e. g. 3 log 4 (5 x + 6) = 2 baseanswer = number inside 42 = 16 = 10 = 5 5 x + 6 5 x x 2 = x

e. g. 4 log 3 (2 x - 4) = 1 + log 3 (4 x + 8) log 3 (2 x – 4) – log 3 (4 x + 8) = 1 For an unknown power always take logs of both sides e. g. 5 6 n = 3200 log 10 6 n = log 10 3200 n log 10 6 = log 10 3200 n = log 10 3200 log 10 6 12 x + 24 = 2 x - 4 10 x = -28 10 x = -2. 8 n = 4. 5045

Calculations using log 10 1000 = 3 as 103 = 1000 If we want log 2 32 = log 10 32 logn m = loga m loga n change of base law log 10 2 e. g. 6 log 2 55 = log 10 x log 10 2 5. 78 = log 10 x 105. 78 = x x = 604449 baseanswer = number inside