Higher Outcome 3 Higher Unit 3 Exponential Growth & Decay Special “e” and Links between Log and Exp Rules for Logs Exam Type Questions Solving Exponential Equations Experimental & Theory

Higher Outcome 3 The Exponential & Logarithmic Functions Exponential Graph Logarithmic Graph y x y x (0,1) (1,0)

Higher Outcome 3 Joan puts £2500 into a savings account earning 13% interest per annum. How much money will she have if she leaves it there for 15 years? Let £A(n) be the amount in her account after n years, then: Exponential Growth & Decay Example

Higher Outcome 3 The population of an urban district is decreasing at the rate of 2% per year. (a) Taking P 0 as the initial population, find a formula for the population, P n, after n years. (b) How long will it take for the population to drop from to ? Exponential Growth & Decay Example

Higher Outcome 3 ? Suggests that Exponential Growth & Decay

Higher Outcome 3 Using our formula And setting up the graphing calculator In the fifth year the population drops below Exponential Growth & Decay

Higher Outcome 3 Let P 0 be the initial population Set : The rabbit population on an island increases by 15% each year. How many years will it take for the population to at least double? After 4 years the population doubles Exponential Growth & Decay Example

Higher Outcome 3 The letter e represents the value 2.718….. (a never ending decimal). This number occurs often in nature f(x) = x = e x is called the exponential function to the base e. A Special Exponential Function – the “Number” e

Higher Outcome 3 The mass of a fixed quantity of radioactive substance decays according to the formula m = 50e -0.02t, where m is the mass in grams and t is the time in years. What is the mass after 12 years? A Special Exponential Function – the “Number” e Example

Higher Outcome 3 In chapter 2.2 we found that the exponential function has an inverse function, called the logarithmic function. The log function is the inverse of the exponential function, so it ‘undoes’ the exponential function: Linking the Exponential and the Logarithmic Function

Higher Outcome Linking the Exponential and the Logarithmic Function

Higher Outcome Examples (a)log 3 81 = “ to what power gives ?” (b)log 4 2 = “ to what power gives ?” (c)log 3 =“ to what power gives ?” Linking the Exponential and the Logarithmic Function

Higher Outcome 3 Rules of Logarithms

Higher Outcome 3 Examples Simplify: a)log log b)log 3 63 – log 3 7 Rules of Logarithms Since

Higher Outcome 3 Example Since Rules of Logarithms Since

Higher Outcome 3 You have 2 logarithm buttons on your calculator: which stands for log 10 which stands for log e log ln Try finding log on your calculator 2 Using your Calculator and its inverse and its inverse

Higher Outcome 3 Solve 5 1 = 5 and 5 2 = 25 so we can see that x liesbetween 1 and 2 Taking logs of both sides and applying the rules Solving Exponential Equations Since Example

Higher Outcome 3 For the formula P(t) = 50e -2t : a)Evaluate P(0) b)For what value of t is P(t) = ½P(0)? Solving Exponential Equations (a) Remember a 0 always equals 1 Example

Higher Outcome 3 For the formula P(t) = 50e -2t : b)For what value of t is P(t) = ½P(0)? Solving Exponential Equations ln = log e e log e e = 1 Example

Higher Outcome 3 The formula A = A 0 e -kt gives the amount of a radioactive substance after time t minutes. After 4 minutes 50g is reduced to 45g. (a) Find the value of k to two significant figures. (b) How long does it take for the substance to reduce to half it original weight? Example (a) Solving Exponential Equations

Higher Outcome 3 (a) Solving Exponential Equations Example

Higher Outcome 3 Solving Exponential Equations ln = log e e log e e = 1 Example

Higher Outcome 3 (b) How long does it take for the substance to reduce to half it original weight? Solving Exponential Equations ln = log e e log e e = 1 Example

Higher Outcome 3 When conducting an experiment scientists may analyse the data to find if a formula connecting the variables exists. Data from an experiment may result in a graph of the form shown in the diagram, indicating exponential growth. A graph such as this implies a formula of the type y = kx n Experiment and Theory y x

Higher Outcome 3 We can find this formula by using logarithms: If Then So Compare this to So Is the equation of a straight line Experiment and Theory log y log x (0,log k)

Higher Outcome 3 Experiment and Theory From We see by taking logs that we can reduce this problem to a straight line problem where: And log y log x (0,log k) YmXc=+ Y Xcm

Higher Outcome 3 Example The following data was collected during an experiment: a) Show that y and x are related by the formula y = kx n. b) Find the values of k and n and state the formula that connects x and y. X y Experiment and Theory

Higher Outcome 3 a)Taking logs of x and y and plotting points we get: Since we get a straight line the formula connecting X and Y is of the form X y

Higher Outcome 3 b) Since the points lie on a straight line, formula is of the form: Graph has equation Compare this to Experiment and Theory Selecting 2 points on the graph and substituting them into the straight line equation we get:

Higher Outcome 3 Sub in B to find value of c Experiment and Theory Sim. Equations Solving we get The two points picked are and ( any will do ! )

Higher Outcome 3 So we have Compare this to andso Experiment and Theory

Higher Outcome 3 solving so You can always check this on your graphics calculator Experiment and Theory

Revision Logarithms & Exponentials Higher Mathematics Next

Logarithms Revision Back Next Quit Reminder All the questions on this topic will depend upon you knowing and being able to use, some very basic rules and facts. Click to show When you see this button click for more information

Logarithms Revision Back Next Quit Three Rules of logs

Logarithms Revision Back Next Quit Two special logarithms

Logarithms Revision Back Next Quit Relationship between log and exponential

Logarithms Revision Back Next Quit Graph of the exponential function

Logarithms Revision Back Next Quit Graph of the logarithmic function

Logarithms Revision Back Next Quit Related functions of Move graph left a units Move graph right a units Reflect in x axis Reflect in y axis Move graph up a units Move graph down a units Click to show

Logarithms Revision Back Next Quit Calculator keys lnln = l og =

Logarithms Revision Back Next Quit Calculator keys lnln = 2.5= = 0.916… l og = 7.6= = … Click to show

Logarithms Revision Back Next Quit Solving exponential equations Show Take log e both sides Use log ab = log a + log b Use log a x = x log a Use log a a = 1

Logarithms Revision Back Next Quit Solving exponential equations Take log e both sides Use log ab = log a + log b Use log a x = x log a Use log a a = 1 Show

Logarithms Revision Back Next Quit Solving logarithmic equations Change to exponential form Show

Logarithms Revision Back Next Quit Simplify expressing your answer in the form where A, B and C are whole numbers. Show

Logarithms Revision Back Next Quit Simplify Show

Logarithms Revision Back Next Quit Find x if Show

Logarithms Revision Back Next Quit Givenfind algebraically the value of x. Show

Logarithms Revision Back Next Quit Find the x co-ordinate of the point where the graph of the curve with equation intersects the x -axis. When y = 0 Exponential form Re-arrange Show

Logarithms Revision Back Next Quit The graph illustrates the law If the straight line passes through A(0.5, 0) and B(0, 1). Find the values of k and n. Gradient y-intercept Show

is the area covered by the fire when it was first detected and A is the area covered by the fire t hours later. If it takes one and a half hours for the area of the forest fire to double, find the value of the constant k. Logarithms Revision Back Next Quit Before a forest fire was brought under control, the spread of fire was described by a law of the form where Show

Logarithms Revision Back Next Quit The results of an experiment give rise to the graph shown. a)Write down the equation of the line in terms of P and Q. It is given that and stating the values of a and b. b) Show that p and q satisfy a relationship of the form Gradient y-intercept Show

Logarithms Revision Back Next Quit The diagram shows part of the graph of. Determine the values of a and b. Use (7, 1) Use (3, 0) Hence, from (2) and from (1) Show

Logarithms Revision Back Next Quit The diagram shows a sketch of part of the graph of a)State the values of a and b. b)Sketch the graph of Graph moves 1 unit to the left and 3 units down Show

Logarithms Revision Back Next Quit a) i) Sketch the graph of ii) On the same diagram, sketch the graph of b)Prove that the graphs intersect at a point where the x-coordinate is Show

Logarithms Revision Back Next Quit Part of the graph of is shown in the diagram. This graph crosses the x-axis at the point A and the straight line at the point B. Find algebraically the x co-ordinates of A and B. Show

Logarithms Revision Back Next Quit The diagram is a sketch of part of the graph of a)If (1, t ) and ( u, 1) lie on this curve, write down the values of t and u. b)Make a copy of this diagram and on it sketch the graph of c)Find the co-ordinates of the point of intersection of with the line a) b) c) Show

Logarithms Revision Back Next Quit The diagram shows part of the graph with equation and the straight line with equation These graphs intersect at P. Solve algebraically the equation and hence write down, correct to 3 decimal places, the co-ordinates of P. Show

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