Maths for A Level Physics - updated edition

Example C: Justify, by calculation, the assertion that the gradient of e x is equal to the value of e x . Consider the two points, x = 1 and x = 2. We’ll calculate the mean gradients within ± 0.1 of these values of x . The table shows the signiϐicant values of e x . x 0.9 1.0 1.1 1.9 2.0 2.1 e x 2.460 2.718 3.004 6.686 7.389 8.166 Consider x = 1: e x = 2.718. Mean gradient around this point = 3.004 – 2.460 0.2 = 2.72 Consider x = 2: e x = 7.389. Mean gradient around this point = 8.166 – 6.686 0.2 = 7.40 In both cases the values of gradient are within 0.15% of the value of e x which is a reasonable justiϐication. It is suggested that the reader repeat these calculations for other values of x and for smaller ranges, e.g. ± 0.01. 4.4 Logarithms 4.4.1 What are logarithms? We start with two uncontroversial statements: 64 = 2 6 and 64 = 4 3 . What relationship do the 6 and the 3 have to 64? They are, of course, the powers to which we have to raise 2 and 4, in order to get 64. In this context, the 2 and the 4 are called bases . The 6 is called the logarithm of 64 to the base 2. We write 6 = log 2  64. The 3 is called the logarithm of 64 to the base 4: 3 = log 4  64 Thus, if X = b p then, by deϐinition, p = log b X . Example D: 243 = 3 5 and 64 = 3 3.7856 (approx.). Express these relationships in terms of logarithms. (a) 243 = 3 5 ∴ 5 = log 3  243. (b) 64 = 3 3.7856 ∴ 3.7856 = log 3  64. In what follows, we shall assume that the base is greater than 1. Note these two fairly obvious identities. Whatever the value of b : log b  1 = 0 (because b 0 = 1) and log b b = 1 (because b 1 = b ). Also log b  0 does not exist, because we’d have to ϐind a number x such that b x = 0. Similarly we cannot have log b  (−1) or any other negative number. 4.4.2 Popular bases for logarithms There are only two popular bases: 10 and e (the natural number 2.71828…). Other bases are seldom used in Physics – except in books like this, to help you to understand what logarithms are all about! log 10 X is often written simply as log  X . ‘log’ is probably what you’ll ϐind on your calculator button. Base 10 logarithms have the nice feature that log 10  0.1 = −1, log 10  1 = 0, log 10  10 = 1, etc. Sound levels in decibels (dB) and star ‘brightnesses’ in magnitudes are both deϐined in terms of the base 10 logarithms of certain ratios of measurable quantities. log e X is usually written as ln  X . ‘ln’ signiϐies natural logarithm . Natural logarithms are used a great deal in Physics. You’ll see why later on. QUICKFIRE 4.13 Find: (a) log 3 81 (b) log 81 3 QUICKFIRE 4.14 Evaluate (a) log 3 √ 3 (b) log 3 9 √ 3 (c) log 3 1 3 (d) log 3 1 √ 3 QUICKFIRE 4.15 81 = 27 4 3 . What is log 27 81? There’s no such thing (in the realm of real numbers) as the logarithm of a negative number, e.g. what is log 2 (−8)? [If tempted by −3, remember that 2 −3 = 1 8 .] 4 Indices and Logarithms 39