By Hobson A.J.
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This booklet offers over 500 functional feedback designed to aid tutors determine lively studying among their scholars. Divided into necessary sections the ideas hide the total variety of training and studying occasions and contain a 'start anywhere', dip-in source compatible for either the newcomer and the previous hand.
Advent. ch 1: the fundamentals of Algebra. ch 2: Rational and Irrational Numbers. ch three: Two-Dimensional Figures. ch four: Linear services. ch five: Powers. ch 6: the fundamentals of facts. ch 7: three-d Geometry. ch eight: Proportional Reasoning and Percents. 258 pages
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We have simply replaced the y in the statement x = B y by logB x in the equivalent statement y = logB x. (b) For any number y, y = logB B y . 2 In other words, any number can be expressed in the form of a logarithm without necessarily using a calculator. We have simply replaced x in the statement y = logB x by B y in the equivalent statement x = By . 4 PROPERTIES OF LOGARITHMS The following properties were once necessary for performing numerical calculations before electronic calculators came into use.
3 3. 4 4. 5 5. 5 √ √ √ √ √ 64 = 4 since 43 = 64. −64 = −4 since (−4)3 = −64. 81 = 3 since 34 = 81. 32 = 2 since 25 = 32. −32 = −2 since (−2)5 = −32. Note: If the index of the radical is an odd number, then the radicand may be positive or negative; but if the index of the radical is an even number, then the radicand may not be negative since no even power of a negative number will ever give a negative result. (a) Rules for Square Roots In preparation for work which will follow in the next section, we list here the standard rules for square roots: √ (i) ( a)2 = a √ (ii) a2 = |a| √ √ √ (iii) ab = a b (iv) a b = √ √a b assuming that all of the radicals can be evaluated.
From Result (a) of the previous section, p B logB p = log q = B logB p−logB q , q B B by elementary properties of indices. The result therefore follows. 3 (c) The Logarithm of an Exponential logB pn = n logB p, where n need not be an integer. Proof: From Result (a) of the previous section, pn = B logB p n = B n logB p , by elementary properties of indices. (d) The Logarithm of a Reciprocal logB 1 = − logB q. q Proof: This property may be proved in two ways as follows: Method 1. The left-hand side = logB 1 − logB q = 0 − logB q = − logB q.
Just the maths - units for students by Hobson A.J.