The Exponential & Logarithmic Functions

2. The Exponential & Logarithmic Functions

3. Exponential Growth & Decay • Worked Example: • 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:

4. ? • Example 1: • The population of an urban district is decreasing at the rate of 2% per year. • (a) Taking P0 as the initial population, find a formula for • the population, Pn, after n years. • (b) How long will it take for the population to drop from • 900 000 to 800 000? (a) Suggests that

5. Using our formula And setting up the graphing calculator In the fifth year the population drops below 800 000

6. Let be the initial population Set • Example 2: • 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

7. f(x) = 2.718..x = ex is called the exponential function to the base e. A Special Exponential Function – the “Number” e The letter e represents the value 2.718….. (a never ending decimal). This number occurs often in nature Although we now think of logarithms as the exponents to which one must raise the base to get the required number, this is a modern way of thinking. In 1624 e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work. Certainly by 1661 Huygens understood the relation between the rectangular hyperbola and the logarithm. He examined explicitly the relation between the area under the rectangular hyperbola yx = 1 and the logarithm. Of course, the number e is such that the area under the rectangular hyperbola from 1 to e is equal to 1. This is the property that makes e the base of natural logarithms, but this was not understood by mathematicians at this time, although they were slowly approaching such an understanding

8. Example 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?

9. Linking the Exponential Function and the Logarithmic Function 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:

10. (c)log3 = “…. to what power gives ….?” 2 3 4 • Example 4: (a)log381 = “…. to what power gives ….?” 4 3 81 (b)log42 = “…. to what power gives ….?” 4 2 -3 3

11. Rules of Logarithms Rules for indices: Rules for Logs:

12. Since Since Example 5: Simplify: • b) log363 – log37 a) log102 + log10500

13. Since Since c)

14. log ln Using your Calculator You have 2 logarithm buttons on your calculator: log which stands for log10 and its inverse ln which stands for loge and its inverse Try finding log10100on your calculator 2

15. Solve Solving Exponential Equations 51 = 5 and 52 = 25 so we can see that x lies between ……and………….. 1 2 Taking logs of both sides and applying the rules

16. For the formula P(t) = 50e-2t: • a) evaluate P(0) • b) for what value of t is P(t) = ½P(0)? a) b) Could we have known this?

18. Example • The formula A = A0e-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? (a) Take logs of both sides

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20. (b) How long does it take for the substance to reduce to half it original weight?

21. Experiment and Theory • 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 = kxn

22. So We can find this formula by using logarithms: If Then So Compare this to Is the equation of a straight line

23. So From We see by taking logs that we can reduce this problem to a straight line problem where: And

24. Worked Example: The following data was collected during an experiment: a) Show that y and x are related by the formula y = kxn . b) Find the values of k and n and state the formula that connects x and y.

25. a) Taking logs of x and y gives: 1.69 2.28 2.70 2.84 1.32 1.67 1.92 2.00 Plotting these points We get a straight line and hence the formula connecting X and Y is of the form

26. b)Since the points lie on a straight line, we can say that: If Then So Compare this to

27. By selecting points on the graph and substituting into this equation we get using So Subtract

28. So we have Compare this to You can always check this on your graphics calculator so and solving so