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Powers and Logs

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In addition we have. In multiplication we use powers (also known as indices or exponents) for ... We write loge as ln for natural log. Examples. log3813/4=3/4 4=3 ... – PowerPoint PPT presentation

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Title: Powers and Logs


1
Powers and Logs
  • Rules for Powers
  • Power and Fraction Examples
  • Rules for Logs
  • Compound Interest and Growth Factors
  • Continuously compounded interest

2
Powers and LogsRules for Powers
  • In addition we have
  • In multiplication we use powers (also known as
    indices or exponents) for repeated
    multiplication. We use the notation
  •  
  • We say that 7 is the index or exponent and 3 is
    the base.

3
Powers and LogsRules for Powers
  • Rule
  • am?anamn
  • amnamn
  • (ab)mambm
  • (a/b)mam/bm
  • Example
  • 32 ?33352439 ?27
  • x ?x3x1 ?x3x4
  • (1.05)10 ?1.05121.0522
  • (22)3266443
  • (y2)4y8
  • (1.072)31.076
  • (42 ?52)20240016 ?25
  • x3y3(xy)3
  • 1.053 ? 1.073(1.05 ? 1.07)31.12353
  • 1.053 ? 1.073(1.05 ?1.07)30.9813

4
Powers and LogsRules for Powers Fractional
Powers
  • If aa1a1/2 ?a1/2
  • Then a1/2 must be that number when multiplied by
    itself gives a. i.e a1/2?a
  • Similarly a1/3 3?a which we call the cube root
    of a
  • In general a1/m m?a and
  • an/m (m?a)n
  • Examples
  • 90.5?93
  • (90.5)2 (?9)29
  • 84/3(3?8)42416
  • x4/5x2/5x6/5(5?x)6
  • ?(x4?25) ?x4??25 5x2

5
Powers and LogsRules for PowersNegative Powers
  • If a2a3/aa3 ?a-1
  • Then a-1 must be 1/a
  • Similarly a-2 1/a2
  • In general a-m1/am
  • Examples
  • 4-21/421/16
  • 4-0.51/40.51/2
  • a2/a-1a3
  • (a2b-3c-5)/(a-1bc2)a3/b4c7

6
Powers and Logs Power and Fraction Examples
  • Power and Fraction examples
  • More Problems.doc

7
Powers and LogsRules for Logs
  • Logs are powers in reverse
  • If amb then logabm
  • Examples
  • log4643
  • log5252
  • log661
  • log3661/20.5
  • log2731/3
  • loga01 (for any a)
  • log100.1-1
  • log101022

8
Powers and LogsRules for Logs
  • Rules
  • logb(xy)logbxlogby
  • logb(x/y)logbx-logby
  • logbxyylogbx
  • logaxlogbx/logba
  • logaaxx
  • A special base is base e, where e is known as
    Eulers constant. e2.71818. We write loge as ln
    for natural log
  • Examples
  • log3813/43/4?43
  • log1025log1040log10(25?40) log1010003
  • log12821/7
  • lne33
  • log612log63log6(12 ? 3)2
  • log exercises.doc

9
Powers and LogsRules for Logs
  • Application
  • If SP(1r)T, where Pinitial investment
    Sfuture of the investment rinterest rate
    (expressed as a decimal) and Ttime, how many
    years does it take for an investment to double if
    it is compounding at 5 p.a.?
  • Investment doubles ? S2P
  • (1r)T2
  • ln(1r)Tln2?T?ln(1r)ln2?Tln2/ln(1.05)14.2
  • 14.2 years
  • How would the answer change if we used log10 not
    ln ?

10
Powers and LogsRules for Logs
  • Rule of 72. At a compounding interest rate of
    r?100 p.a. it takes approximately 72/(r?100)
    years to double the value of an investment. Where
    does this rule come from?
  • Use your calculator to calculate the following
  • ln(1.1)0.095
  • ln(1.05)0.0488
  • ln(1.01)0.00995
  • ln(1.005)0.004988
  • ln(1.001)0.0009995
  • If x ix small what is the approximate value of
    ln(1x)?
  • General formula for doubling time is
    Tln2/ln(1r).
  • r is a little more than ln(1r)
  • We need to replace ln(2)0.69, by a number which
    is a little greater than 0.69 eg. 0.72.

11
Powers and LogsRules for Logs
  • If r0.07 how long does it take to double your
    investment?
  • Exact formula gives
  • Tln(2)/ln(1.07)10.24
  • Rule of 72 gives
  • T72/710.28
  • Repeat the above if r0.15 and r0.01
  • r0.15 T4.96 (exact formula) T4.8 (72 rule)
  • r0.01 T69.7 (exact formula) T72 (72 rule)

12
Powers and LogsLogs Exact Rule vs Rule of 72
13
Powers and LogsCompound Interest and Growth
Factors
  • If I invest 1 for 1 year at a rate of 10 p.a
    compound interest how much will I have at the end
    of one year?
  • 1?1.11.10
  • At the end of 2 years?
  • 1?1.1 ?1.1 1?1.121.21
  • At the end of k years?
  • 1?1.1k
  • If the interest rate was 5 how would the above
    formula change?
  • 1?1.05k
  • If the interest was was R (or rR/100)?
  • 1?(1r)k
  • If I invested P at R p.a. for k years?

14
Powers and LogsCompound Interest and Growth
Factors
  • Compound interest growth formula
  • SP(1r)T
  • Sfinal value of investment
  • PPrinciple invested
  • rinterest rate expressed as a decimal
  • Ttime
  • r and T must be in the same units of time
  • If r is interest p.a. then T is years
  • If r is weekly interest rate then T is in weeks
  • If T is quarters then r is a quarterly interest
    rate

15
Powers and LogsCompound Interest and Growth
Factors
  • Growth Factor. The amount by which we multiply
    the principle P to get the final value of the
    investment S is called then growth factor F
  • For ordinary compound interest
  • F(1r)T

16
Powers and LogsCompound Interest and Growth
Factors
  • Examples
  • If F1.055 for an investment of 1 year what was
    the interest rate
  • F(1r)1? rF-10.0555.5
  • If F1.006 for an investment of 1 month, what is
    r
  • Expressed as interest per month?
  • r0.0060.6 p.m
  • Expressed as interest per year?
  • Growth factor for one year is (1.006)121.0744
  • r0.07447.44 p.a.

17
Powers and LogsCompound Interest and Growth
Factors
  • Examples
  • If the growth factor for one year is F1.07, what
    is the quarterly interest rate?
  • F(1r)T. To get a quarterly interest rate we
    must express T in quarters. 1 year 4 quarters,
    therefore T4.
  • F(1r)4 1rF1/4 rF1/4-11.070.25-10.017
  • r1.7 p.quarter
  • If an investment has grown from 25000 to 30000
    over 30 months what is the annual compound
    interest rate?

18
Compound Interest and Growth Factors
  • Example
  • If an investment has grown from 25000 to 30000
    over 30 months what is the annual compound
    interest rate?
  • FS/P(1r)T30000/250001.2
  • 30 months 2.5 years T2.5
  • 1.2(1r)2.5 r1.22/5-10.07567.56

19
Compound Interest and Growth Factors
  • Converting Interest rates into different time
    periods
  • Is 1 monthly compound interest equal to 12 per
    annum?
  • No
  • Growth factor over 1 year F(1r)T1.01121.1268
  • Interest rate over one year is 12.68
  • What is 8p.a. expressed as a quarterly compound
    interest rate?
  • r1.081/4-10.01941.94
  • What is 8p.a. expressed as a monthly compound
    interest rate?
  • r1.081/12-10.00640.64
  • What is 8p.a. expressed as a weekly compound
    interest rate?
  • r1.081/52-10.00150.15

20
Compound Interest and Growth Factors Converting
Interest rates into different time periods
  • Where do these formula come from?
  • By noting that the growth factor for the for the
    same time period must be equal irrespective of
    how the interest is expressed
  • Growth Factor for 1 year
  • F(1ra) ra is annual
  • Growth Factor for 1 year
  • F(1rw)52, rw is weekly
  • (1ra)(1rw)52
  • ra(1rw)52-1
  • rw(1 ra)1/52-1

21
Powers and LogsCompound Interest and Growth
Factors Converting Interest rates into different
time periods
  • In general if the year is divided into k periods
    then
  • (1rk)k1ra where ra is the annual rate and rk
    is the interest rate for one kth of the year
  • This gives
  • rk(1ra)1/k-1
  • ra(1rk)k-1

22
Compound Interest and Growth Factors Converting
Interest rates into different time periods
  • Examples
  • If you are quoted a quarterly interest rate of
    4.25, what is the annual interest rate?
  • ra(1rk)k-1(1.0425)4-10.181118.11
  • If you are quoted an annual rate of 12.5 how
    much is the monthly interest rate?
  • rw(1ra)1/k-11.1251/12-10.009680.968
  • If you invest 5000 at an interest rate of 2 per
    quarter. How much is the investment worth after 7
    months?

23
Powers and LogsCompound Interest and Growth
Factors Converting Interest rates into different
time periods
  • If you invest 1 at an interest rate of 2 per
    quarter, what is the growth factor after 7
    months?
  • F(1r)T if r is a quarterly interest rate then
    T must also be expressed in quarters
  • T7/3 quarters
  • F(1.02)7/31.048
  • An investment has grown from 12,500 to 15,373
    over 2 years. What is the return expressed as a
    weekly interest rate?
  • F15375/125001.23
  • F(1r)T both T and r must be in the same units.
    T104 weeks
  • r1.231/104-10.0020.2 per week

24
Powers and LogsCompound Interest and Growth
Factors
  • An investment has grown from 4000 to 4200 in 30
    days. If this growth is sustained over a year
    how much will the investment be worth at the end
    of the year?
  • Growth factor for 30 days is 4200/40001.05
  • 30 day return 5
  • Growth factor for one year is (1ra) where ra is
    the return for 1 year 365 days)
  • 1ra(1.05)365/301.81
  • SP(1ra)T4000?(1.81)7242

25
Powers and LogsCompound Interest and Growth
Factors
  • On 31st Dec 1989 the All Ordinaries Index was
    1655.
  • On 31st Dec 1999 All Ordinaries Index was 3141.
  • If the All Ordinaries Index is a measure of the
    value of the Australian share market, what was
    the growth in the Australian share market
    expressed as an annual rate of return?
  • Growth factor for 10 years F3141/16551.898
  • F(1ra)T T10 years ra1.8980.1-10.06626.62

26
Powers and LogsCompound Interest and Growth
Factors
  • You invest 5000 over 31/2 years and saw a
    growth factor of 1.6
  • What is the per annum growth rate?
  • 1.6(1ra)3.5 ra1.61/3.5-10.143714.37
  • What was the monthly growth rate
  • 3.5 years 42 months
  • 1.6(1rm)42 rm1.61/42-10.01121.12

27
Powers and LogsContinuously Compounded Interest
  • Banks often quote interest in the following way
  • 12 per annum paid monthly
  • Translation You pay 1 per month
  • We know (1.01)12?1.12
  • (1.01)121.1268, so you are actually paying
    12.68
  • Bank speak 12 per annum paid weekly
  • Translation You pay 12/52 per month
  • Actual rate is (1.12/52)12-10.1273
  • What happens as the time period gets smaller and
    smaller?

28
Powers and LogsContinuously Compounded
Interest12 paid per time period
29
Powers and LogsContinuously Compounded
InterestThe magic number e
30
Powers and LogsContinuously Compounded
InterestContinuous vs Ordinary Compounded
Interest
  • Ordinary ro
  • SP(1ro)T
  • Growth Factor
  • S/P(1ro)T
  • Continuous rc
  • SPercT
  • Growth Factor
  • S/PercT

31
Powers and LogsContinuously Compounded Interest
  • Examples
  • You are offered an investment at 4.5 p.a. over 4
    years. If you have 11,000 to invest and the
    contract specifies continuously compounded
    interest, how much do you receive at the end of 4
    years?
  • SPercT P11,000 rc0.045 T4 years
  • S13,169.39
  • What was the monthly continuously compounded rate
    for this investment?
  • SPercT S13,169.39,P11,000 rc ? T48months
  • rcln(S/P)?1/T0.003750.375
  • Notice that the monthly rate annual rate/12

32
Powers and LogsContinuously Compounded Interest
  • Example
  • If interest is quoted as 12 continuously
    compounding what is the
  • Quarterly rate?
  • 0.12/40.033
  • Monthly rate?
  • 0.12/120.011
  • Weekly?
  • 0.12/520.00230.23

33
Powers and LogsContinuously Compounded Interest
  • Suppose you have 22,500 to invest and you are
    offered the following continuously compounded
    interest rates
  • 6.5 for the 1st year
  • 6.0 for the 2nd year
  • 5.5 for the third year
  • What is the value of the final investment?
  • S22,500?e0.065 ?e0.06 ?e0.055
    22500?e0.0650.0600.055
  • S 22,500?e0.1826,937.39
  • What is the continuously compounded interest rate
    for the entire 3 years?
  • r3years0.18
  • What is the average annual interest rate?
  • ra0.18/30.066

34
Powers and LogsContinuously Compounded Interest
  • Repeat the previous example using ordinary
    compounding interest
  • What is the value of the final investment?
  • S22,500?(1.065) ?(1.06)?(1.055)22,500?1.191
  • S26,797.26
  • What is the ordinary compounded interest rate for
    the entire 3 years?
  • (1r3years)1.191 r0.19119.1
  • What is the average annual interest rate?
  • (1ra)31.191 ra1.1911/3-10.066

35
Continuously Compounded Interest
  • Which investment offers the highest return
  • 7.5 ordinary p.a.
  • 7.2 continuous p.a.
  • 1.8 ordinary per quarter
  • 0.61 continuous per month
  • To compare we need to convert all the interest
    rates to a common base. Choose continuous p.a.
  • 7.5 Ordinary rcln(10.75)0.07237.23
  • 7.2 continuous p.a.
  • 1.8 ordinary per quarter is (1.018)4-1
    0.0747.4 ordinary p.a
  • 0.61 continuous per month12?0.00610.07327.32
  • Highest return is 1.8 ordinary per quarter
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