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© Boardworks Ltd 2005 1 of 26 © Boardworks Ltd 2005 1 of 26 AS-Level Maths: Core 2 for Edexcel C2.6 Exponentials and logarithms This icon indicates the slide contains activities created in Flash. These activities are not editable. For more detailed instructions, see the Getting Started presentation.

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© Boardworks Ltd 2005 2 of 26 Contents © Boardworks Ltd 2005 2 of 26 Exponential functions Logarithms The laws of logarithms Solving equations using logarithms Examination-style questions Exponential functions

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© Boardworks Ltd 2005 3 of 26 Exponential functions So far in this course we have looked at many functions involving terms in x n. In an exponential function, however, the variable is in the index. For example: The general form of an exponential function to the base a is: y = a x where a > 0 and a ≠1. You have probably heard of exponential increase and decrease or exponential growth and decay. A quantity that changes exponentially either increases or decreases more and more rapidly as time goes on. y = 2 x y = 5 x y = 0.1 x y = 3 – x y = 7 x +1

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© Boardworks Ltd 2005 4 of 26 Graphs of exponential functions

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© Boardworks Ltd 2005 5 of 26 Exponential functions In both cases the graph passes through (0, 1) and (1, a ). This is because: a 0 = 1 and a 1 = a for all a > 0. When 0 < a < 1 the graph of y = a x has the following shape: y x 1 1 When a > 1 the graph of y = a x has the following shape: y x (1, a )

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© Boardworks Ltd 2005 6 of 26 Contents © Boardworks Ltd 2005 6 of 26 Exponential functions Logarithms The laws of logarithms Solving equations using logarithms Examination-style questions Logarithms

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© Boardworks Ltd 2005 7 of 26 Logarithms Find p if p 3 = 343. We can solve this equation by finding the cube root of 343: Now, consider the following equation: Find q if 3 q = 343. We need to find the power of 3 that gives 343. One way to tackle this is by trial and improvement. Use the x y key on your calculator to find q to 2 decimal places.

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© Boardworks Ltd 2005 8 of 26 Logarithms To avoid using trial and improvement we need to define the power y to which a given base a must be raised to equal a given number x. This is defined as: y = log a x “ y is equal to the logarithm, to the base a, of x ” This can be written using the implication sign : y = log a x a y = x The expressions y = log a xa y = x andare interchangeable. For example, 2 5 = 32 can be written in logarithmic form as: log 2 32 = 5

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© Boardworks Ltd 2005 9 of 26 Logarithms Taking a log and raising to a power are inverse operations. We have that: y = log a x a y = x So: Also: y = log a a y For example: 2and6

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© Boardworks Ltd 2005 10 of 26 Contents © Boardworks Ltd 2005 10 of 26 Exponential functions Logarithms The laws of logarithms Solving equations using logarithms Examination-style questions The laws of logarithms

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© Boardworks Ltd 2005 11 of 26 Some important results When studying indices we found the following important results: This can be written in logarithmic form as: log a a = 1 a1 = aa1 = a This can be written in logarithmic form as: log a 1 = 0 a0 = 1a0 = 1 It is important to remember these results when manipulating logarithms.

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© Boardworks Ltd 2005 12 of 26 The laws of logarithms The laws of logarithms follow from the laws of indices: The multiplication law Let: m = log a x and n = log a y So: x = a m and y = a n log a x + log a y = log a ( xy ) xy = a m × a n Using the multiplication law for indices: xy = a m + n Writing this in log form gives: m + n = log a xy But m = log a x and n = log a y so:

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© Boardworks Ltd 2005 13 of 26 The laws of logarithms The division law Let: m = log a x and n = log a y So: x = a m and y = a n Using the division law for indices: Writing this in log form gives: But m = log a x and n = log a y so:

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© Boardworks Ltd 2005 14 of 26 The laws of logarithms The power law Let: m = log a x So: x = a m Using the power law for indices: Writing this in log form gives: But m = log a x so: x n =( a m ) n x n = a mn mn = log a x n n log a x = log a x n

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© Boardworks Ltd 2005 15 of 26 The laws of logarithms These three laws can be used to combine several logarithms written to the same base. For example: Express 2log a 3 + log a 2 – 2log a 6 as a single logarithm.

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© Boardworks Ltd 2005 16 of 26 The laws of logarithms Express log 10 in terms of log 10 a, log 10 b and log 10 c. Logarithms to the base 10 are usually written as log or lg. We can therefore write this expression as: The laws of logarithms can also be used to break down a single logarithm. For example:

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© Boardworks Ltd 2005 17 of 26 Logarithms to the base 10 and to the base e Although the base of a logarithm can be any positive number, there are only two bases that are commonly used. These are: Logarithms to the base 10 Logarithms to the base e Logarithms to the base 10 are useful because our number system is based on powers of 10. They can be found by using the log key on a calculator. Logarithms to the base e are called Napierian or natural logarithms and have many applications in maths and science. They can be found by using the ln key on a calculator.

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© Boardworks Ltd 2005 18 of 26 Changing the base of a logarithm Suppose we wish to calculate the value of log 5 8. We can’t calculate this directly using a calculator because it only find logs to the base 10 or the base e. We can change the base of the logarithm as follows: Let x = log 5 8 So:5 x = 8 Taking the log to the base 10 of both sides: log 5 x = log 8 x log 5 = log 8 So: 1.29 (to 3 s.f.)

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© Boardworks Ltd 2005 19 of 26 Changing the base of a logarithm If we had used log to the base e instead we would have had: In general, to find log a b : Let x = log a b, so we can write a x = b Taking the log to the base c of both sides gives: log c a x = log c b x log c a = log c b 1.29 (to 3 s.f.) So:

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© Boardworks Ltd 2005 20 of 26 Contents © Boardworks Ltd 2005 20 of 26 Exponential functions Logarithms The laws of logarithms Solving equations using logarithms Examination-style questions Solving equations using logarithms

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© Boardworks Ltd 2005 21 of 26 Solving equations involving logarithms We can use the laws of logarithms to solve equations. For example: Solve log 5 x + 2 = log 5 10. To solve this equation we have to write the constant value 2 in logarithmic form: 2 = 2 log 5 5 because log 5 5 = 1 = log 5 5 2 = log 5 25 The equation can now be written as: log 5 x + log 5 25 = log 5 10 log 5 25 x = log 5 10 25 x = 10 x = 0.4

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© Boardworks Ltd 2005 22 of 26 Solving equations of the form a x = b We can use logarithms to solve equations of the form a x = b. For example: Find x to 3 significant figures if 5 2 x = 30. We can solve this by taking logs of both sides: log 5 2 x = log 30 2 x log 5 = log 30 Using a calculator: x = 1.06 (to 3 s.f.)

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© Boardworks Ltd 2005 23 of 26 Solving equations of the form a x = b Find x to 3 significant figures if 4 3 x +1 = 7 x +2. Taking logs of both sides:

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© Boardworks Ltd 2005 24 of 26 Solving equations of the form a x = b Solve 3 2 x –5(3 x ) + 4 = 0 to 3 significant figures. If we let y = 3 x we can write the equation as: So: If 3 x = 1 then x = 0. Now, solving 3 x = 4 by taking logs of both sides:

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© Boardworks Ltd 2005 25 of 26 Contents © Boardworks Ltd 2005 25 of 26 Exponential functions Logarithms The laws of logarithms Solving equations using logarithms Examination-style questions

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© Boardworks Ltd 2005 26 of 26 Examination-style question Julia starts a new job on a salary of £15 000 per annum. She is promised that her salary will increase by 4.5% at the end of each year. If she stays in the same job how long will it be before she earns more than double her starting salary? 15 000 × 1.045 n = 30 000 1.045 n = 2 log 1.045 n = log 2 n log 1.045 = log 2 15.7 Julia’s starting salary will have doubled after 16 years.

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