@@ -5,10 +5,10 @@ title: 2.6 Logarithms
55
66* a logarithm is simply an inverse exponential function
77* b^x=y => x=log(y) where the base of the logarithm is b
8- * exponential grows really fast, inverse exponential (log) grows slowly
8+ * exponential grows really fast, inverse exponential (log) grows slowly
99* logarithms arise in any process where things are repeatedly halved
1010
11- ## Applications
11+ ## Applications of logarithms
1212
1313### binary search
1414
@@ -46,8 +46,26 @@ title: 2.6 Logarithms
4646
4747### summations
4848
49- * Harmonic numbers: special case of arithmetic progression
49+ * Harmonic numbers: special case of arithmetic progression
5050* H(n) = S(n, -1)
5151* the sum of the progression of simple reciprocals
5252
5353![ image] ( images/2.6-harmonic_numbers.jpg )
54+ 55+ ## Properties of logarithms
56+ 57+ * important bases: log2 (lg), loge (ln), log10(log)
58+ * loga(xy) = loga(x) + loga(y)
59+ * loga(n^b) = b * loga(n)
60+ * changing the base: loga(b) = logc(b) / logc(a)
61+ 62+ ### consequences
63+ 64+ * ** the base of the logarithm has no real impact on the growth rate**
65+ * (since changing the base of the log from a to c involves dividing by logc(a) and that is lost to the Big Oh notation whenever a and c are constants)
66+ * ** logarithms cut any function down to size**
67+ * the growth rate of the logarithm of ANY polynomial function is O(lg n)
68+ * log(n!) = &Theta ; (n log n)
69+ 70+ 71+
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