Education Resources Logs and Exponentials Higher Mathematics Supplementary Resources Section A This section is designed to provide examples which develop routine skills necessary for completion of this section. R1 I have had experience of simplifying expressions with surds and indices. 1. Simplify the following writing the answers with positive indices only. (c) 2. (c) (g) (h) (i) R2 I can write an exponential in logarithmic form and vice versa. 1. For each exponential relationship, write a related logarithmic relationship. (c) (g) (h) (i) (j) (k) (l) (m) (n) (o) Higher Mathematics Logs and Exponentials Page 1
2. For each logarithmic relationship, write a related exponential relationship. (c) (g) (h) (i) (j) (k) (l) R3 I can solve exponential equations using logarithms. 1. Solve each of the following exponential equations (c) (g) (h) (i) (j) (k) (l) 2. Solve each of the following exponential equations (c) (g) (h) 4 (i) (j) 3 (k) (l) 3. Solve each of the following exponential equations (c) SLC Education Resources Biggar High School Page 2
Section B This section is designed to provide examples which develop Course Assessment level skills NR1 I can manipulate logarithms and exponentials and apply the three main laws of logarithms. 1. Given which of the following is true: 2. Given which of the following is true: 3. Simplify 4. Show that.. 5. If find an expression for x in terms of y and z. 6. Find a if. 7. Simplify expressing your answer in the form where and are whole numbers. Higher Mathematics Logs and Exponentials Page 3
NR2 I can solve exponentials and logarithmic equations using the laws of logarithms. 1. Given the equation find the value of when giving your answer to 3 significant figures. 2. Given the equation, find, to 3 significant figures: when, and. when, and. (c) when, and. 3. Solve the equation. Solve. (c) Solve. 4. Given that, show that. Solve. 5. The curve with equation, where, cuts the x-axis at the point. Find the value of. 6. If, find the value of. 7. Solve the equation. 8. Find if. 9. Find the x-coordinate of the point where the graph of the curve with equation intersects the x-axis. SLC Education Resources Biggar High School Page 4
NR3 I have experience of plotting and extracting information from straight line graphs with logarithmic axes (axis). 1. Show that, where and are constants, can be expressed as a straight line in terms of and. 2. Show that, where and are constants, can be expressed as a straight line in terms of and. 3. Variables and are related by the equation The graph of against is a straight line through the points and, as shown in the diagram. Find the values of and. 4. Two variables, and, are connected by the law. The graph of against is a straight line passing through the origin and the point. Find the value of. 5. 2 The graph illustrates the law. 1 If the straight line passes through and, find the values of and. Higher Mathematics Logs and Exponentials Page 5
6. The result of an experiment gives rise to the graph shown. Write down the equation of the line in terms of and. If and, show that and satisfy stating the values of and. -2 2 1 P Q 7. Variables and are related by the equation The graph of against is a straight line through the points and, as shown in the diagram. Find the values of and. 8. Two variables and satisfy the equation. Find the value of if lies on the graph with equation. If also lies on the graph, find. (c) A graph is drawn of against. Show that its equations will be of the form and state the gradient of this line. SLC Education Resources Biggar High School Page 6
NR4 I can solve logarithmic and exponential equations in real life contexts. 1. The amount of a certain radioactive element,, remaining after years can be found using the formula, where is the amount present initially. If 300 grams are left after 500 years, how many grams were present initially. The half-life of a substance is the time taken for the amount to decrease to half its initial amount. What is the half life of this substance? 2. The value (in thousand) of a car is shown to depreciate after years from first purchase according to the formula. What was the value of the car when first purchased? The car was sold when its value had dropped to 10% of the value when first purchased. After how many years was the car sold? 3. The formula is used to determine the age of wood, where is the amount of carbon-14 in any living tree, is the amount of carbon-14 in the wood being dated and is the age of the wood in years. A wooden artefact was found to contain 90% of the carbon-14 of a living tree. Is the artefact over 500 years old? Higher Mathematics Logs and Exponentials Page 7
4. The size of a rabbit population,, can be modelled using the equation where is the population at the beginning of a study and t is the time in years since the study began and is a constant. The rabbit population comprised of 70 individuals at the beginning of the study. If find the size of the rabbit population after six years. How long will it take the rabbit population to double in size? 5. Radium decays exponentially and its half-life is 1600 years. If represents the amount of radium in a sample to start with and represents the amount remaining after years, then. Determine the value of, correct to 3 significant figures. Hence find what percentage, to the nearest whole number, of the original amount of radium will be remaining after 2500 years. 6. The concentration of a fertiliser in the soil can be modelled by the equation where is the initial concentration, is the concentration at time and is the time, in days, after the application of the pesticide. If it takes 20 days for the level of the fertiliser in the soil to reduce by 25%, find the value of to 2 significant figures. Eighty days after the initial application, what is the percentage decrease in the concentration of the fertiliser? 7. The spread of disease in trees was described by a law of the form where is the area covered by the disease when it was first detected and is the area covered by the disease months later. If it takes six months for the area of the disease to double, find the value of the constant,, correct to 3 significant figures. SLC Education Resources Biggar High School Page 8
NR5 I can display on, and extract information from, logarithmic and exponential graphs. 1. The diagram shows the curves with equations and. The graphs intersect at the point A. Show that the x-coordinate of can be written in the form, for all. Calculate the y-coordinate of. 2. The diagram shows the graph of,. On separate diagrams sketch: (c) 3. Sketch the graph of. On the same diagram sketch: (c) Higher Mathematics Logs and Exponentials Page 9
4. Sketch the graph of. On the same diagram sketch the graph of. Prove that the two graphs intersect at a point where the x-coordinate is. 5. The diagram shows a sketch of part of the graph of Write down the values of and. Sketch the graph of 6. The function is of the form. The graph of the diagram. is shown in Write down the values of and. State the domain of. SLC Education Resources Biggar High School Page 10
NR6 I have experience of cross topic exam standard questions Logarithms with composite functions 1. Functions, and are defined on suitable domains by, and. Find expressions for and. Hence solve. Logarithms with polynomials 2. Show that is a root of. Hence factorise fully. Solve. Higher Mathematics Logs and Exponentials Page 11
Answers R1 1. (c) 2. (c) (g) (h) (i) R2 1. a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) 2. a) b) c) d) e) f) g) h) i) j) k) l) R3 (Answers to 3 significant figures where appropriate) 1. a) b) c) d) e) f) g) h) i) j) k) l) SLC Education Resources Biggar High School Page 12
2. a) b) c) d) e) f) g) h) i) j) k) l) 3. a) b) c) d) e) f) NR1 1. 2. 3. 3. 4. Proof 4. Proof 5. 6. 7. NR2 1. 2. 2. 2(c). 3. and 3. 3(c). and 4. Proof 4. 5. 6. 7. 8. 9. NR3 1. against 2. against 3. and 4. 5. and 6. 6. and 7. and 8. 8. 8(c). Proof and gradient = NR4 1. grams 1. years 2. 2. years 3. The artefact is 850 years old which is over 500 years old. Higher Mathematics Logs and Exponentials Page 13
4. rabbits 4. years 5. 5. 6. 6. 7. NR5 1. 1. 2. 2. 2(c). 3. SLC Education Resources Biggar High School Page 14
4. 4. Proof 5. and 5. 6. and 6. Domain = NR6 1. and 1. and 2. 2. Higher Mathematics Logs and Exponentials Page 15