Chpt 4 - Logarithms - Original

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) + (xlog = xlog : x of valuepositive the Find 42

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-5 Example 14 0 = 3 + 3log 4- xlog : x of valuethe Find x3 TUTORIAL 4 1. Write each of the following in logarithmic form. For example, 3 4 = 81 can be written as log3 81 = 4. )3 1( = 81 (d) 16 = 64 (c) 5 = 125 (b) 16 = 2 (a) -4 2 334 2. Write each of the fo llowing in exponential form. For example, log 5 125 = 3 can be written as 53 = 125. (a) log232 = 5 (b) 2 = log525 (c) 7 = log2128 (d) -2 = log3(1/9) (e) log e1 = 0 (f) 2 = logaX (g) ln 20.09 = 3 3. Determine the value of each of the following logarithms. 6 10527 7 102 10log)e(125log (d) 3log (c) 10log (b) 64log (a) − 4. Write each of the following as a single logarithm. (a) 3loga 2 + 2loga 3 - 2loga 6 (b) 3log2 5 - 2log2 7 (c) )4x(log27log3 164log2 1 2 555 +−+ (d) )8x(log2x8log)2x(log3 222 +−++ (e) )4x(log)1x2(log3xlog2 555 −++−

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-6 5. Evaluate, without using calculator: (a) 3log102 + 2log105 – log1020 (b) 5log22 41log70log35 41log 10101010 +−+ (c) 50 49log20 21log15 14log 101010 −+ 6. Solve the equations: 16 1 = )3 2( (e) 6 = )2 1( (d) 10 = 22 (c) 4 = 3 (b) 2 = 3 (a) xx 1)+(xx4xx 7. Given that log 2 3 = 1.585, and log2 5 = 2.322, calculate the values of log2 60 and log2 0.3. 8. If log72 = 0.356 and log73 = 0.565, find the value of 2 9log29 8log 77 + 9. Solve each of the following equations. (a) 3)2x(logxlog 22 =++ (b) 2)3x2(logxlog 33 −=+− 10. Find the values of x in (a) log2 x + logx 2 = 2 (b) log3 x - 2logx 3 = 1

LOGARITHMS Mathematics Preparatory Course for Direct Entry to Higher Nitec 4-7 Challenging Questions 1 If ,log4 xu = find in term of u (a) x (b) x2log4 (c) 64logx 2 (a) If px =8log , express x2log in terms ofp . Given that ( ) 3log =xyq and ( ) 4log 32 =yxq . Calculate the values of xqlog and yqlog 3 (a) Calculate the value of 8log3 . Giving your answer correct to 3 significant figures. (b) Evaluate x if ( ) ( ) ( ) 32log5log1log 222 =−−−++ xxx