1. 49: A Practical Application of Log Laws © Christine Crisp “Teach A Level Maths” Vol. 1: AS Core Modules

2. More Laws of Logs "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" Module C2 MEI/OCR

3. More Laws of Logs If, from an experiment, we have a set of values of x and y that we think may be related we often plot them on a graph. If the relationship can be approximated by a straight line, a line of best fit can easily be drawn through the data. However, it is not easy to draw a curve through data. or If we think that a relationship of the form fits the data, where a and b are constants, we can use logs to obtain a straight line.

4. More Laws of Logs Method: Suppose we believe a relationship of the form exists between x and y. Then, Take logs: Simplify: ( Law 1 ) This equation now represents a straight line where ( Law 3 )

5. More Laws of Logs Method: Suppose we believe a relationship of the form exists between x and y. Then, Take logs: Simplify: ( Law 1 ) This equation now represents a straight line where ( Law 3 )

6. More Laws of Logs Method: Suppose we believe a relationship of the form exists between x and y. Then, Take logs: Simplify: ( Law 1 ) This equation now represents a straight line where ( Law 3 )

7. More Laws of Logs Method: Suppose we believe a relationship of the form exists between x and y. Then, Take logs: Simplify: ( Law 1 ) This equation now represents a straight line where ( Law 3 ) and

8. More Laws of Logs Method: Suppose we believe a relationship of the form exists between x and y. Then, Take logs: Simplify: ( Law 1 ) This equation now represents a straight line where ( Law 3 ) and

9. More Laws of Logs e.g. 1 It is believed that the following data may represent a relationship between x and y of the form. Draw a suitable straight line graph to confirm this and estimate the values of a and n. x2.53.14.355.97.18.1 y9152635435769 We have seen that so, to get the straight line we need to plot a graph of against.

10. More Laws of Logs x2.53.14.355.97.18.1 y9152635435769 log x 0.40.490.630.70.770.850.91 log y 0.951.181.411.541.631.761.84 Using logs to base 10 we get so the graph is as follows:

11. More Laws of Logs The constant, c, cannot be read off the graph because the intercept on the y -axis is not shown. From the graph, the gradient, Instead, we substitute the coordinates of any point on the graph ( not from the table ). e.g.

12. More Laws of Logs We now have and So, and We finally need to find a so we must get rid of the log. This is called anti-logging. On the calculator, the anti-log usually shares a button with the log. For base 10 it is marked. ( 2 s.f. )

13. More Laws of Logs so,( as before ) For a relationship of the form we work in a similar way. Take logs:

14. More Laws of Logs so,( as before ) but For a relationship of the form we work in a similar way. Take logs:

15. More Laws of Logs so,( as before ) but For a relationship of the form we work in a similar way. Take logs: The gradient, m =

16. More Laws of Logs so,( as before ) but For a relationship of the form we work in a similar way. Take logs: The gradient, m = and c =

17. More Laws of Logs so,( as before ) but We plot against x. For a relationship of the form we work in a similar way. Take logs: The gradient, m = and c =

18. More Laws of Logs SUMMARY  The relationships and can both be reduced to straight lines by taking logs. For,

19. More Laws of Logs Exercise 1. Use a suitable straight line graph to show that the data fit a relationship of the form x 51525303540 y 2.46.316.326.242.267.9 Estimate the values of a and b to 2 s.f. 2. Explain how you would use a graph to estimate the values of a and n for a set of x and y data thought to fit a relationship of the form

20. More Laws of Logs Solutions 1. Plot against x. x51525303540 log y0.380.801.211.421.631.83 (2 s.f.)

21. More Laws of Logs 2. Convert the equation, by taking logs, to get Read off the value where the line meets the Y -axis to find c OR substitute a pair of ( X, Y ) values into. Calculate values of and. Plot a graph of against. Measure the gradient, m, of the graph to obtain n. Draw the line of best fit. Use and antilog to find a. Solutions

22. More Laws of Logs

23. The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

24. More Laws of Logs SUMMARY  The relationships and can both be reduced to straight lines by taking logs. For, Plot against

25. More Laws of Logs e.g. 1 It is believed that the following data may represent a relationship between x and y of the form. Draw a suitable straight line graph to confirm this and estimate the values of a and n. 6957433526159y 8.17.15.954.33.12.5x We have seen that so, to get the straight line we need to plot a graph of against.

26. More Laws of Logs 6957433526159y 8.17.15.954.33.12.5x 1.841.761.631.541.411.180.95 log y 0.910.850.770.70.630.490.4 log x Using logs to base 10 we get so the graph is as follows:

27. More Laws of Logs The constant, c, cannot be read off the graph because the intercept on the y -axis is not shown. From the graph, the gradient, Instead, we substitute the coordinates of any point on the graph. e.g.

28. More Laws of Logs e.g. Use a suitable straight line graph to show that the data fit a relationship of the form 67.942.226.216.36.32.4 y 40353025155 x Estimate the values of a and b to 2 s.f. Plot against x. 1.831.631.421.210.800.38log y 40353025155x

29. More Laws of Logs (2 s.f.) bxaylog 