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Click to Start. Higher Maths. Unit 3 Chapter 3. Logarithms. Experiment & Theory. Introduction. In experimental work. we often want to create a mathematical model. data can often be modelled by equations. of the form:. Polynomial function. Exponential function. Exponential function.

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  1. Click to Start

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

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

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

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

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

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

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

  9. y 2.3 2.2 2.1 2.0 x 1.0 1.1 1.2 1.3 1.4 1.5 1.6 Example The table shows the result of an experiment How are x and y related ? x x A quick sketch x x suggests x x

  10. Now take logarithms of x and y 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:

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

  12. 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

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

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

  15. When log10y is plotted against log10x, a best fitting straight line Qu. 3 has gradient 2 and passes through the point (0.6, 0.4) Fit this data to the model Given Using c = -0.8 m = 2 n = 2 Relation between x and y is:

  16. When log10y is plotted against log10x, a best fitting straight line Qu. 4 has gradient -1 and passes through the point (0.9, 0.2) Fit this data to the model Given Using c = 1.1 m = -1 n = -1 Relation between x and y is:

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

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

  19. 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 )

  20. 2000  Paper II B11. The results of an experiment give riseto 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 )

  21. THE END

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