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Transcript of Exponents and Logarithms - inside- For Economists/4...  Inside ECONOMICS Mathematics for...

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    Inside ECONOMICS

    Mathematics for Economists

    Exponents and Logarithms Introduction

    Many types of data can be modelled using exponential functions. Exponential functions naturally model the growth of a variable and are functions in which the variable appears as the exponent. For instance a financial variable that grows over time may have time as the exponent, that is () = 2. In this document we investigate the derivative of an exponential function. In finding the derivative of an exponential function we will discover that it describes the inverse of the exponential function which in turn is a logarithm. Understanding exponents and logarithms is crucial for working in applied economics as we often use logarithms to turn multiplicative relationships into additive relationships which are easier to manipulate. When working with data that exhibits exponential growth we often find that the graph is not a perfectly exponential due to volatility between data points. (This can occur for a number of reasons). However, many computer packages have curve fitting tools which allow us to easily develop equations that approximate data.

    Exponential Functions

    Properties of Exponents 1. if is positive integer then is multiplied by itself time 2. = + 3. = 4. () = 5. 0 = 1 6. = 1 (if is a negative number then, is 1 || , the reciprocal of ||)

    7. 1 = an (the nth root of )

    8. = an

    m (the mth power of the nth root of )

    Exponential Growth Functions

    The graph is the function () = = 2

    x y-5 0.03125-4 0.0625-3 0.125-2 0.25-1 0.50 11 22 43 84 165 32

    Exponential Functions

    If is a real number with > 0 and 1, then the function () = is an exponential function with base .

    = 2

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    Notice that the axis is the horizontal asymptote for the graph. However unlike a rational function the graph approaches this asymptote in only a single direction. In the other direction the graph increases more rapidly than a polynomial. These graphs approach the axis but never actually ouch it. The steepness of the graph increases at an exponential rate. The larger the base the quicker the graph becomes asymptotic to the axis in one direction and steep in the other direction.

    x a = 1.5 a = 2 a = 4 a = 6-5 0.131687 0.03125 0.000977 0.000129-4 0.197531 0.0625 0.003906 0.000772-3 0.296296 0.125 0.015625 0.00463-2 0.444444 0.25 0.0625 0.027778-1 0.666667 0.5 0.25 0.1666670 1 1 1 11 1.5 2 4 62 2.25 4 16 363 3.375 8 64 2164 5.0625 16 256 12965 7.59375 32 1024 7776

    = 1.5

    = 2 = 4 = 6

    = () = () ( > 1) Exponential Growth Functions

    The graph represents the general form of exponential growth functions. Note that in general terms the intercept is . If we just have , where = 1, the intercept is 1. Likewise if = 2 then the intercept will become 2.

    (0,)

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    Inside ECONOMICS

    Example 1: Exponential Growth Functions

    If $5000 is invested at 8%, compounded monthly, then what is the future value of the investment after 5 years and 10 years ? Graph the function.

    Firstly we divide 8% by = 12 months for the rate. The apply the following formula, where is the principle amount in year 1, is the future value in year and is the interest rate. So is the total number of compounding periods for which interest is applied. In this case there will be 60 periods for 5 years and 120 periods for 10 years.

    = ( )

    5 = 5000(1.00667)12(5) = $7,449

    10 = 5000(1.00667)12(10) = $11,098

    It is obvious from the graph and formula that compounding interest is essentially a exponential growth function.

    Exponential Decay Functions

    The graph is the function () = = 12

    = 2

    Exponentials whose bases are between zero and one have graphs different to those of exponential growth functions. If a function is of the form = where 0 < < 1 then using the properties of exponents we have

    = = 1

    = ()1 = =

    x a = 0.5-5 32-4 16-3 8-2 4-1 20 11 0.52 0.253 0.1254 0.06255 0.03125

    = 12

    = ( )

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    We have written = in the form = , where > 1 and = . Therefore if we have = (1 2 ), then

    = 12

    = (2)1 = 2

    From the diagrams you can probably notice that the graphs are reflections. For instance the

    graph of = 12 is the reflection of = 2 in the y axis. Likewise = 1

    3 is the reflection

    of = 3 and so forth.

    x a = 1.5 a = 2 a = 3 a = 6-5 7.59375 32 243 7776-4 5.0625 16 81 1296-3 3.375 8 27 216-2 2.25 4 9 36-1 1.5 2 3 60 1 1 1 11 0.666667 0.5 0.333333 0.1666672 0.444444 0.25 0.111111 0.0277783 0.296296 0.125 0.037037 0.004634 0.197531 0.0625 0.012346 0.0007725 0.131687 0.03125 0.004115 0.000129

    = () = () ( > 1)

    Exponential Decay Functions

    The graph represents the general form of exponential decay functions. Again, note that in general terms the intercept is . If we just have , where = 1, the intercept is 1. Likewise if = 2 then the intercept will become 2.

    = 1.5

    = 2 = 4 = 6

    (0,)

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    A Note on Negative Bases

    Negative bases are not valid for exponential functions. A negative integer for would take positive values for when is even and negative values for when is odd. However the function never takes on the value zero in between these values. In addition the square root of a negative number is not even defined. Essentially we only work with exponential functions.

    Example 2: Evaluating Exponents

    Evaluate the exponential function 2 at = 2 and = 3

    2(2) = 4

    2(3) = 8

    By using points between = 2 to = 3 attempt to evaluate every 10 decimal increment between 2 and 3. Explain what you notice. (Hint: it is best to do this in excel)

    As discussed above between even and odd numbers a function with a negative base will not be invalid and is not defined. So for = 2 we have 4 a positive number and for = 3 we have a negative number 8, yet it is never zero in between. This is why we are not allowed to have negative bases for the exponential function.

    Example 3: Exponents

    State whether each of the following statements is either true or false.

    (a) can give a negative value for if = and > 1? (b)The graph of = , with > 1, approaches the positive axis as an asymptote.

    (c) The graph of = 13 is the graph of = 3 reflected about the axis.

    (d) The intercept of the function = 2(3) is given at the point (0,2).

    Solution

    (a) False (the axis is the horizontal asymptote for the graph and therefore > 0) (b) True (the asymptote is the axis in the positive half) (c) True (the graph of these two functions are symmetrical) (d) True = () = ()

    x a y2 -2 4

    2.1 -2 #NUM!2.2 -2 #NUM!2.3 -2 #NUM!2.4 -2 #NUM!2.5 -2 #NUM!2.6 -2 #NUM!2.7 -2 #NUM!2.8 -2 #NUM!2.9 -2 #NUM!3 -2 -8

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    The Number

    In this section we shall discuss the irrational number . The exponential functions with base often arise in natural ways. To understand the irrational number we will examine the growth of a savings account from making an initial investment. Suppose that we deposit $ dollars into a savings account that pays a simple annual interest rate of . This amount will grow to + = (1 + ) in year 1. After two years this amount will be

    (1 + )(1 + ) = (1 + )2

    dollar in the account. Likewise After years the there will be

    (1 + )

    dollars in the account. Now think of the situation when we are compounding more frequently, such as semi annual, quarterly and monthly. For quarterly we are compounding four times per year so we the interest applied in each quarter is /4. After the first quarter the account will have +

    4 = (1 +

    4). After the second quarter we will have (1 +

    4)2

    and after the fourth quarter (1 + 4)4. Therefore the amount of dollars in the account after t

    years we be (1 + 4)4.

    Generally, if interest is compounded times per year there will be (1 +

    )1 after one period

    of compounding, (1 +

    ) after years of compounding.

    Continuously Compounding

    We know that banks and financial institutions offer rates with various compounding such as fortnightly, daily and continuously. If interest is compounded so frequently we are concerned with the limit of (1 +

    ) as approaches infinity. For the purpose of intuition we can

    calculate an interest rate of 100% for different frequencies of compounding.

    The tables shows an increasing sequence in which converges to approximately 2.718. It turns out that this number is irrational and cannot be written as a fraction or a repeating decimal. We use the letter to denote this number. In addition this number plays an important role in economics and finance. We call the following function the exponential function.

    () =

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    Graph of Exponential Function

    () =

    () =

    Example 4: Exponential Functions

    Suppose a financial institution offers continuously compounded interest rates on deposits of $100 or more. How much will $100 be at 5% per annum continuously compounded for 3, 5 and 10 years?

    x e^x-7 0.001-6 0.002-5 0.007-4 0.018-3 0.050-2 0.135-1 0.3680 11 32 73 204 555 1486 4037 1097

    x e^x-7 1096.6-6 403.4-5 148.4-4 54.6-3 20.1-2 7.4-1 2.70 1.01 0.3682 0.1353 0.0504 0.0185 0.0076 0.0027 0.001

    () =

    lim

    1 +1