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3. Higher Maths Unit 3 Chapter 3 Logarithms Experiment & Theory

4. In experimental work Introduction data can often be modelled by equations of the form: we often want to create a mathematical model Polynomial function Exponential function

5. Polynomial function Exponential function Often it is difficult to know which model to choose

6. A useful way is to take logarithms (i)For This is like or rearranging

7. A useful way is to take logarithms (ii)For This is like or rearranging

8. Plotting log y against log x gives us a straight line In the case of a simple polynomial In the case of An exponential Plotting log y against x gives us a straight line

9. By drawing the straight line graph The constants for the gradient m the y-intercept c can be found cm c m

10. 1.01.6 2.0 2.3 x y 2.1 2.2 1.11.21.31.41.5 Example The table shows the result of an experiment x1.11.21.31.41.51.6 y2.062.112.162.212.262.30 How are x and y related ? A quick sketch x x x x x x suggests

11. Now take logarithms of x and y 0.040.080.110.150.180.20 0.310.320.330.340.350.36 We can now draw the best fitting straight line Take 2 points on the line (0.04, 0.31) and (0.18, 0.35) To find c, use:

12. Recall our initial suggestion Taking logs of both sides n = 0.29 a = 2 (1 dp) Our model is approximately

13. Putting it into practice 1.From the Graph find the gradient 2.From the Graph find or calculate the y-intercept Make sure the graph shows the origin, if reading it directly 3.Take logs of both sides of suggested function 4.Arrange into form of a straight line 5.Compare gradients and y-intercept

14. Qu. 1 AssumeExpress equation in logarithmic form From the graph: m = 0.7 c = 0.2 Relation between x and y is: Find the relation between x and y n = 0.7

15. Qu. 2 AssumeExpress equation in logarithmic form From the graph: m = -0.7 c = 0.4 Relation between x and y is: Find the relation between x and y n = -0.7

16. Qu. 3 When log 10 y is plotted against log 10 x, a best fitting straight line Given m = 2 c = -0.8 Relation between x and y is: has gradient 2 and passes through the point (0.6, 0.4) n = 2 Fit this data to the model Using

17. Qu. 4 When log 10 y is plotted against log 10 x, a best fitting straight line Given m = -1 c = 1.1 Relation between x and y is: has gradient -1 and passes through the point (0.9, 0.2) n = -1 Fit this data to the model Using

18. Qu. 5 AssumeExpress equation in logarithmic form From the graph: m = 0.0025 c = 0.15 Relation between x and y is: Find the relation between x and y

19. Qu. 6 AssumeExpress equation in logarithmic form From the graph: m = -0.015 c = 0.6 Relation between x and y is: Find the relation between x and y

20. 2002  Paper I 11.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. ( 4 )

21. 2000  Paper II B11. 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. ( 2 ) It is given that and b)Show that p and q satisfy a relationship of the form stating the values of a and b. ( 4 )

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