Exponential and Logarithmic Functions

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1 Eponential and Logarithmic Functions Eponential functions are those with variable powers, e.g. = a. Their graphs take two forms: (0, 1) (0, 1) When a > 1, the graph: is alwas increasing is alwas positive never cuts the ais passes through (0, 1) shows eponential growth Eponential Functions as Models When 0 < a < 1, the graph is alwas decreasing is alwas positive never cuts the ais passes through (0, 1) shows eponential deca Eample 11: Ulanda s population in 2016 was 100 million and it was growing at 6% per annum. a) Find a formula P n for the population in millions, n ears later. b) Estimate the population in the ear 2026 Eample 12: 8000 gallons of oil are lost in an oil spill in Blue Sk Ba. At the beginning of each week a filter plant removes 67% of the oil present. a) Find a formula G n for the amount of oil left in the ba after n weeks. b) After how man complete weeks will there be less than 10 gallons left? I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 96 of 107

2 (0, 1) (1, 2) = 2 Logarithmic Functions The inverse of an eponential function is known as a logarithmic function. If f() = a, then f -1 () = log a ( log to the base a of ) We have seen that the graph of the inverse of a function can be obtained b reflection in the line =. Since the graph of = 2 passes through the points (0, 1) and (1, 2), then the inverse of f() = 2 must pass through the points (1, 0) and (2, 1). Eample 13: Add the graph of = log 2 to the graph opposite. Note that: = a passes through (0, 1) and (1, a) = log a passes through (1, 0) and (a, 1) = a means a multiplied b itself times gives = log a means is the number of times I multipl a b itself to get Since the graph does not cross the -ais, we can onl take the logarithm of a positive number The epression log a can be read as a to the power of what is equal to?, e.g. log 2 8 means 2 to the power of what equals 8?, so log 2 8 = 3. Eample 14: Write in logarithmic form: a) 5 2 = 25 b) 12 1 = 12 c) = 2 d) 8 = e) 1 = q 0 f) ( 3) 4 = k Eample 15: Write in eponential form: a) 3 = log b) log 7 49 = 2 c) log = 6 1 d) log e) log b g = 5h f) 1 = log 7 7 Eample 16: Evaluate: a) log 8 64 b) log 2 32 c) log d) log e) log 4 2 Since a 1 = a, then log aa = 1 Since a 0 = 1, then log a 1 = 0. I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 97 of 107

3 Laws of Logarithms log a = log a + log a log a = log a - log a log a n = n log a Eample 17: 1 a) log log 2 8 log 2 2 b) 2log 5 10 log c) Simplif (log3 810 log 3 10) 4 Solving Logarithmic Equations You MUST memorise the laws of logarithms to solve log equations! As we can onl take logs of positive numbers, we must remember to discard an answers which violate this rule! Eample 18: Solve: a) log 4 (3 2) log 4 ( + 1) = b) log 6 + log 6 (2-1) = The Eponential Function and Natural Logarithms The graph of the derived function of = a can be plotted and compared with the original function. The new graphs are also eponential functions. Below are the graphs of = 2 and = 3 (solid lines) and their derived functions (dotted). f ( ) = 2 f ( ) = 3 The derived function of = 2 lies under the original graph, but the derived function of = 3 lies above it. This means that there must be a value of a between 2 and 3 where the derived function lies on the original. i.e. where f() = f () The value of the base of this function is known as e, and is approimatel I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 98 of 107

4 The function = e is known as The Eponential Function. The function = log e is known as the Natural Logarithm of, and is also written as ln. Eample 19: Evaluate: Eample 20: Solve: a) e 3 b) log e 120 a) ln = 5 b) 5-1 = 16 Eample 21: Atmospheric pressure P t at various heights above sea level can be determined b using the rt formula Pt P0 e, where P 0 is the pressure at sea level, t is the height above sea level in thousands of feet, and r is a constant. a) At feet, the air pressure is half that at sea level. Find r accurate to 3 significant figures. b) Find the height at which P is 10% of that at sea level. Eample 22: A radioactive element decas according to the law A t = A 0e kt, where A t is the number of radioactive nuclei present at time t ears and A 0 is the initial amount of radioactive nuclei. a) After 150 ears, 240g of this material had decaed to 200g. Find the value of k accurate to 3 s.f. b) The half-life of the element is the time taken half the mass to deca. Find the half-life of the material. I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 99 of 107

5 Using Logs to Analse Data, Tpe 1: = k n log = n log + log k When the data obtained from an eperiment results in an eponential graph of the form = k n as shown below, we can use the laws of logarithms to find the values of k and n. To begin, take logs of both sides of the eponential equation. = k n log log = k n This gives a straight line graph! log = n log + logk Note: the base is not important, as long as the same base is used on both sides. log Eample 23: Data are recorded from an eperiment and the graph opposite is produced. a) Find the equation of the line in terms of log 10 and log 10. O log 10 b) Hence epress in terms of. Using Logs to Analse Data, Tpe 2: = kn log = log n ( )+ log k A similar technique can be used when the graph is of the form = kn (i.e. is the inde, not the base as before). = kn log = kn log = (log n) + logk I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 100 of 107

6 Eample 24: The data below are plotted and the graph shown is obtained. log log a) Epress log 10 in terms of. O b) Hence epress in terms of. I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 101 of 107

7 Related Graphs of Eponentials and Logs = e + a = ln( + a) (0, 1) (1, 0) = e + a is obtained b sliding = e : = ln( + a) is obtained b sliding = ln Verticall upwards if a>0 Horizontall left if a>0 Verticall downwards if a<0 Horizontall right if a<0 ( + a) = e = k ln (0, 1) (1, 0) = e ( + a) is obtained b sliding = e = k ln is obtained b verticall: Horizontall left if a>0 stretching = ln if k > 1 Horizontall right if a<0 compressing = ln if 0 < k < 1 = e - = - ln (0, 1) (1, 0) = e - is obtained b reflecting = e : = - ln is obtained b reflecting = ln : in the -ais in the - ais I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 102 of 107

8 Eample 25: The graph of = log 4 is shown. On the same diagram, sketch: a) = log 4 4 (4, 1) O (1, 0) b) = log Past Paper Eample 1: a) Show that = 1 is a root of = 0, and hence factorise full b) Solve log 2( + 3) + log 2( ) = 3 Past Paper Eample 2: Variables and are related b the equation = k n. The graph of log 2 against log2 is a straight line through the points (0, 5) and (4, 7), as shown in the diagram. Find the values of k and n. log 2 (0, 5) O (4, 7) log 2 I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 103 of 107

9 Past Paper Eample 3: The concentration of the pesticide Xpesto in soil is modelled b the equation: Pt P 0 is the initial concentration kt P0 e where: P t is the concentration at time t t is the time, in das, after the application of the pesticide. a) Once in the soil, the half-life of a pesticide is the time taken for its concentration to be reduced to one half of its initial value. If the half-life of Xpesto is 25 das, find the value of k to 2 significant figures. b) Eight das after the initial application, what is the percentage decrease in Xpesto? Past Paper Eample 4: Simplif the epression 3log 2e 2log 3e giving our answer in the form e e A logeb logec, where A, B and C are whole numbers. Past Paper Eample 5: Two variables and satisf the equation 34. A graph is drawn of log10 against. Show that its equation will be of the form log10 P Q, and state the gradient and -intercept of this line. I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 104 of 107

10 Logs & Eponentials Trigonometr Sets, Functions & Graphs Vectors Epressions & Functions Unit Topic Checklist: Unit Assessment Topics in Bold Topic Questions Done? Logarithms Eercise 15E, (all) Y/N Laws of logarithms Eercise 15F, Q 1 Y/N Log equations Eercise 15G, Q 1, 2, 3 Y/N e and Natural logarithms Eercise 15D, Q 1 5 Y/N Eponential growth/deca Eercise 15H, Q 4-7 Y/N Data and straight line graphs Eercise 15J, Q 2; Eercise 15K, Q 2 Y/N Related log & eponential graphs Eercise 15K, Q 1-7 Y/N Radians Eercise 4C, Q 1 3 Y/N Eact Values Eercise 4E, Q 1, 3 Y/N Trig Identities Eercise 17, Q 2; Eercise 11J, Q 20 Y/N Compound and double angle formulae Eercise 11D, Q 6 8; Eercise 11F, Q 1 4, 7, 9 Y/N k cos( - ) Eercise 16C, Q 1 5; Eercise 16D, Q 2 Y/N k cos( + ) Eercise 16E, Q 1 Y/N k sin( ) Eercise 16E, Q 2, 3; Eercise 16E, Q 4, 5 Y/N Wave Fn Maima and minima Eercise 16G, Q 1, 3, 4, 5, 7 Y/N Solving Wave Fn equations Eercise 16H, Q 1-4 Y/N Transforming graphs Eercise 3P, Q 1 9 Y/N Naming/Sketching trig graphs Eercise 4B, (all) Y/N Completing the square Eercise 8D, Q 4, 6; Eercise 5, Q 3, 4 Y/N Graphs of derived functions Eercise 6P, (all) Y/N Set Notation Eercise 2A, Q 2 & 3 Y/N Composite Functions Eercise 2C, Q 5-10 Y/N Inverse Functions Eercise 2D, Q 2; Eercise 2I, Q 1 Y/N Graphs of inverse functions Eercise 2F, Q 1 & 2 Y/N Eponential & log graphs Eercise 3N, Q 3, 4; Eercise 3O, p 47, Q 2, 3 Y/N Resultant vectors Eercise 13N (all) Y/N Unit Vectors (inc. i, j, k) Eercise 13F, Q 1, 2; Y/N Collinearit Eercise 13N, Q 15 18, 23 Y/N Section Formula Eercise 13N, Q Y/N Scalar Product Eercise 13O, Q 1; Eercise 13P, Q 1, 2 Y/N Angle between vectors Eercise 13Q, Q 1, 2; Eercise 13S, Q 4 7 Y/N Perpendicular Vectors Eercise 13R, Q 1-8 Y/N Properties of Scalar Product Eercise 13U, Q 1, 2, 4, 5 Y/N I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 105 of 107

11 Functions and Graphs Recurre nce Relation Straight Line Differentiation Integration Past Paper Questions b Topic Topic P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 Ranges and Domains c 15b Composite Functions 3 3 2a,b 1a 1 3a 5b 2a 12a 1 Inverse Functions 5a 6 6 Transforming graphs c Interpreting trig functions and graphs 11b 4a Eact Values 1 13 Terms of Recurrence Relations 4 7a 1 1 3a 3a 9a 8 Creating & using RR formulae Finding a limit of an RR 7b 3c b 3b,c 9b Solving RR s to find a and b 3a,b Gradients from points or equations 2 2 Gradients from angles with - ais 8 4 1c 1b Equations of straight lines a 23b 5 2a 1 1b 11 1b Perpendicular bisectors 21c 23a 1a 1a Altitudes 1b 1a Medians 1a 1b 1a 7 Points of intersection of lines 1c 21b 23c 2b,c 1b 1c 1c 1c Distance Formula 23d Collinearit 9 10a Finding derivatives of functions 6, b Equations of tangents to curves 3a,1 0b 5a Increasing & decreasing functions 21a 9b 15c 4c Stationar points 5 9b 22b 18 21b 9a 7a Curve Sketching 9c 22a c Closed Intervals b Graphs of derived functions Optimisation Velocit, Acceleration, Displacement 9 Finding indefinite integrals 11, Definite Integrals 5 Area under a curve 6 8c 21b 12 3b 15b Area between two curves Differential Equations 5 10b 15 9 I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 106 of 107

12 Quadrat ics Polnomia ls Circles Trigono metr Vectors Past Paper Questions b Topic Topic P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 P1 P2 Completing the square b 12b 15a 4a Quadratic Inequations Roots using b 2 4ac 2 9 1b 3 3b 2 4 Tangenc using b 2 4ac 3b 4 2c, d 22c,d 3b 11b Snthetic Division 9b, c 8b 7 21a 6 3a a 2 Intersections of curves Functions from graphs 10a a Approimate roots (Iteration) ( a) 2 + ( b) 2 = r 2 7 2b 23c 4 4a 10b General equation of a circle 2a a 8 4a Lines cutting circles 2a 23ab Tangents to circles 2b 3, 5c 6 22b 2 11a 8 2 Distance between centres vs radii 4b Trig Equations 7 6 4b 10 6b Compound angle formulae 8b 2a b Double angle formulae 7 8a,b 6 2b ,18 10 Trigonometric identities 7b 11a 11a Interpreting vector diagrams 19 4a 5a Unit Vectors b Position Vectors and Components 2 1 1a, b 10 5ai b 7 5a 5b Collinearit Section Formula 7 11a Scalar Product (angle form) 5b Scalar Product (components) 9a 17 5b 5a Angle between vectors 5aii, 9d 1 1c b 4c 5b 5c Perpendicular Vectors 14 1 Properties of Scalar Product WF Further Calculus Logs and Epone ntials Wave Functions 10a 11a 6a 22a 23 4,9 9 8a 14 Chain Rule (inc. trig functions) 10a, b 11c Integrating a (..) n 14, b 13 2b Integrating sin and cos 9, 7 6b 6 7a,c 5 10b Eponential growth/deca Log equations , Eponential and log graphs a,b 10 Linearisation I Brson. Amendments b E Mawell, M Doran, E Tranor & C Cassells Page 107 of 107

ZETA MATHS. Higher Mathematics Revision Checklist

ZETA MATHS. Higher Mathematics Revision Checklist ZETA MATHS Higher Mathematics Revision Checklist Contents: Epressions & Functions Page Logarithmic & Eponential Functions Addition Formulae. 3 Wave Function.... 4 Graphs of Functions. 5 Sets of Functions