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

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

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.

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 )

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 )

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 )

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

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

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. x y We have seen that so, to get the straight line we need to plot a graph of against.

More Laws of Logs x y log x log y Using logs to base 10 we get so the graph is as follows:

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.

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. )

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

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

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 =

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 =

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 =

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

More Laws of Logs Exercise 1. Use a suitable straight line graph to show that the data fit a relationship of the form x y 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

More Laws of Logs Solutions 1. Plot against x. x log y (2 s.f.)

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

More Laws of Logs

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

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 y x We have seen that so, to get the straight line we need to plot a graph of against.

More Laws of Logs y x log y log x Using logs to base 10 we get so the graph is as follows:

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.

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

More Laws of Logs (2 s.f.) bxaylog 