Maths for A Level Physics - updated edition

Chapter Indices and Logarithms 4.1 Introduction This comparatively short chapter covers two important areas of maths which are used frequently in A-level Physics. AS candidates may need the content of Section 4.2 only – this will depend upon the speciϐication. Questions 1–10 of Test Yourself 4.1 relate to Section 4.2. 4 4.2 Indices a 4 is a compact abbreviation for a × a × a × a in which a can be any number. We pronounce a 4 as ‘ a to the power 4’, or just ‘ a to the 4’. The number 4 is called the exponent or the index. In this context, the plural of index is indices . The number a is the same as a 1 – the index tells you how many a ’s are multiplied together: in this case there is only one a . Indices have a vital role in Physics, for example in expressing very large or very small quantities in standard SI units. Thus, to three ϐigures, the mass of the deuterium (heavy hydrogen) nucleus can be written as 2.38 × 10 −27  kg, the gravitational constant G = 6.67 × 10 −11  kg −1 m 3 s −2 and the solar mass as 1.99 × 10 30  kg (try writing that out in non-indexed form!). It is therefore crucial to be able to perform calculations involving quantities expressed in this way using the laws which follow. 4.2.1 The laws of indices 1. What is the result if we multiply a 4 by a 3 ? a 4 × a 3 = ( a × a × a × a ) × ( a × a × a ) = a × a × a × a × a × a × a = a 7 So, in general, a m × a n = a ( m + n ) Law 1 Thus, in order to multiply two numbers, we add the indices. 2. What if we divide a 4 by a 3 ? a × a × a × a a × a × a = a = a 1 So, in general, a m a n = a ( m − n ) Law 2 3. Finally, what is the third power of a 4 ? ( a 4 ) 3 = ( a × a × a × a ) × ( a × a × a × a ) × ( a × a × a × a ) = a 12 So, in general, ( a m ) n = a mn . Law 3 Example A: Calculate (1.5 3 ) 2 . (1.5 3 ) 2 = 1.5 3 × 2 = 1.5 6 = 11.39 (4 s.f.) 4.2.2 Negative and zero indices We know that a m a n = a ( m − n ) but we can also write a m a n as the product a m × 1 a n . To get the right answer using the ϔirst law of indices, we need to write 1 a n = a − n . a 3 2 means a to the power of 3 2 i.e. a 9 . QUICKFIRE 4.1 Calculate 5 2 × 2 5 QUICKFIRE 4.2 Calculate: (a) (3 4 ) 2 and (b) 3 4 2 . When using laws of indices remember that x can be written as x 1 . 35