# Hobson A.J.'s Just the maths - units for students PDF

By Hobson A.J.

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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.