Basic Engineering Mathematics, Fifth Edition

108 Basic Engineering Mathematics 4x^2 + 6 x− 8 = 0 6 x^2 − 11. 2 x− 1 = 0 3x(x+ 2 )+ 2 x(x− 4 )= 8 4x^2 −x( 2 x+ 5 )= 14 ...

Solving quadratic equations 109 Hence, the mass will reach a height of 16m after 0 .59s on the ascent and after 5.53s on the des ...

110 Basic Engineering Mathematics Ka= x^2 v( 1 −x) , determinex, the degree of ionization, given thatv=10dm^3. A rectangular bu ...

Chapter 15 Logarithms 15.1 Introduction to logarithms With the use of calculators firmly established, logarith- mic tables are n ...

112 Basic Engineering Mathematics Here are some worked problems to help understand- ing of logarithms. Problem 1. Evaluate log 3 ...

Logarithms 113 15.2 Laws of logarithms There are three laws of logarithms, which apply to any base: (1) To multiply two numbers: ...

114 Basic Engineering Mathematics =log ( 4 × 3 25 ) by the first and second laws of logarithms =log ( 12 25 ) =log0. 48 Problem ...

Logarithms 115 Problem 21. Solve the equation log ( x^2 − 3 ) −logx=log2 log ( x^2 − 3 ) −logx=log ( x^2 − 3 x ) from the second ...

116 Basic Engineering Mathematics Problem 22. Solve the equation 2x=5, correct to 4 significant figures Taking logarithms to bas ...

Logarithms 117 y 0.5 0 123 2 0.5 2 1.0 x x 3 0.48 2 0.30 1 0 0.5 2 0.30 0.2 2 0.70 0.1 y 5 log 10 x 2 1.0 Figure 15.1 Hence,loga ...

Chapter 16 Exponential functions 16.1 Introduction to exponential functions An exponential function is one which containsex,e be ...

Exponential functions 119 In problems 3 and 4, evaluate correct to 5 decimal places. (a) 1 7 e^3.^4629 (b) 8. 52 e−^1.^2651 (c ...

120 Basic Engineering Mathematics From equation (1), ex= 1 +x+ x^2 2! + x^3 3! + x^4 4! +··· Hence, e^0.^5 = 1 + 0. 5 + ( 0. 5 ) ...

Exponential functions 121 Table 16.1 x − 3. 0 − 2. 5 − 2. 0 − 1. 5 − 1. 0 − 0. 5 0 0. 5 1. 0 1. 5 2. 0 2. 5 3. 0 ex 0. 05 0. 08 ...

122 Basic Engineering Mathematics A table of values is drawn up as shown below. t 0 1 2 3 e−t/^3 1.00 0.7165 0.5134 0.3679 v= 25 ...

Exponential functions 123 lne^3 x=ln7 i.e. 3 x=ln7 from which x= 1 3 ln7=0.6486, correct to 4 decimal places. Problem 10. Evalua ...

124 Basic Engineering Mathematics Taking natural logs of both sides gives ln 7 4 =lne^3 x ln 7 4 = 3 xlne Sincelne=^1 , ln 7 4 = ...

Exponential functions 125 5= 8 ( 1 −e −x 2 ) ln(x+ 3 )−lnx=ln(x− 1 ) ln(x− 1 )^2 −ln3=ln(x− 1 ) ln(x+ 3 )+ 2 = 12 −ln(x− 2 ) ...

126 Basic Engineering Mathematics When R= 5. 4 × 103 , α= 1. 215477 ...× 10 −^4 and R 0 = 5 × 103 θ= 1 1. 215477 ...× 10 −^4 ln ...

Exponential functions 127 (a) θ 1 , correct to the nearest degree, whenθ 2 is 50 ◦C,tis 30s andτis 60s and (b) the timet, correc ...