Maths for A Level Physics - updated edition

4.4.3 The laws of logarithms These are just the laws of indices written in a different way. All the same, they’re worth learning in the new form… (a) log b XY = log b X + log b Y Law 1 This is equivalent to the ‘add indices’ rule for multiplication and we can derive it from this rule: Suppose X = b p and Y = b q : then by deϐinition log b X = p and log b Y = q . But XY = b p + q , ∴ log b XY = p + q = log b X + log b Y  QED Example E: Given that log 2 = 0.3010, without using your calculator calculate log 20 [Remember log 2 is short for log 10  2, etc.] log 20 = log (2 × 10) = log 2 + log 10 = 0.3010 + 1.0000 =1.3010 Now use your calculator to check this. (b) log b X Y = log b X − log b Y Law 2 This is equivalent to the ‘subtract indices’ rule for division and can be derived in a similar way to the ϐirst law. (c) log b X n = n  log b X Law 3 Suppose X = b m then, by deϐinition, log b X = m . And X n = b mn , so log b X n = mn = n  log b X  QED Example F: ln 100 = 4.605 [4 s.f.]. Without using the ln button, calculate ln 10 . ln 10 = ln 100 1 2 = 1 2  ln 100 = 1 2 × 4.605 = 2.303 Check: Calculator gives ln 10 = 2.303 [4 s.f.] If we put n = −1 into Law 3, we see that log b  1 X = − log b X . We can also derive this result from Law 2 by putting X = 1 and remembering that log b  1 = 0. The following two laws are presented for completeness but you will rarely come across them in A-level Physics. (d) log a X = log a b × log b X Law 4 The derivation is left as an exercise. Hint: start by putting X = b p and b = a q . As an example: ln  X = ln 10 × log  X = 2.303 log  X To make this clearer: log e X = log e  10 × log 10 X = 2.303 log 10 X The 2.303 is, then, the conversion factor between logarithms to base 10 and natural logarithms. (e) log a b = 1 log b a Law 5 Hint: to derive this, put X = a into Law 4. Mathematics for Physics 40