Chpt 4 - Logarithms - With Handwritten notes

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-1 CHAPTER 4 INTRODUCTION The logarithm of Y to the base a is the exponent to which a must be raised to yield Y. That is, xYloga = if and only if Yax = Thus 24log 2 = since 422 = and 38log 2 = since 823 = Example 1 a) 3 2 = 9 ; then 2 = log39 2 is the logarithm of 9 to base 3. b) 102 = 100 ; so 2 = log10100 2 is the logarithm of 100 to base 10. Example 2 If log 10N = 3, find the value of N. Example 3 Given that log x81 = 4, find the value of x. Example 4 Evaluate, log 82 without using calculator.

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-2 4.1 LAWS OF LOGARITHMS A. Product logaxy = logax + logay B. Quotient loga x y = logax - logay C. Power logaxn = nlogax 4.2 SPECIAL LOGARITHMS 1. logaa = 1 e.g. log 22 =1, log55 =1 2. loga1 = 0 e.g. log21= 0, log91= 0 3. loga 1 x = - logax 4. Natural logarithm ln N = logeN (where e = 2.71828183 ) Example 5 25 - 20 + 2Simplify loglog5loglog 3333 + Example 6 3 log - 10 log + 3 log3 Simplify (assume same base) Example 7 4logloglog 222 + 8 - 16 Simplify

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-3 Example 8 Write the following expression as a single logarithm. )5x(log)2x(log3xlog2 2 101010 +−++ Example 9 5 = 2 equation the Solve x Example 10 1.2 of value the find 1.465;=5 0.631,=2 that Given logloglog 333 4.3 CHANGE OF BASE a) Logarithms to base 10 are called "common logarithms", and are denoted by lg. When the base is 10, this number is generally omitted. i.e. log N denotes the logarithm of N to the base 10. b) Logarithms to base e are called "natural logarithms", and are denoted by ln. e has approximately the value 2.718. i.e. log e N is written as ln N

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-4 c) Logarithms can be to any base; however common logarithms are exclusively used for calculations at this stage. d) Where logarithms to other bases are encountered, they have to be changed to base 10 for a numerical answer. To change from base a to base b : a N = N b b a log loglog Example 11 10 log (b) 4 log (a) of value theFind 23 Example 12 2 3 = x log + x log : x of value theFind 42 Example 13 6) +