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KAY174 MATHEMATICS III Prof. Dr. Doğan Nadi Leblebici EXPONENTIAL AND LOGARITHMIC FUNCTIONS PURPOSE: TO UNDERSTAND THE ROLE OF AN EXPONENTIAL FUNCTION IN BUSINESS AND ECONOMICS WHICH INVOLVES A CONSTANT RAISED TO A VARIABLE POWER. EXPONENTIAL AND LOGARITHMIC FUNCTIONS There is a function which has an important role not only in mathematics but in business and economics as well. It involves a constant raised to a variable power and is called an exponential function. An example is f(x) = 2x EXPONENTIAL AND LOGARITHMIC FUNCTIONS The function f defined by y=f(x) = bx Where b>0, b≠1 and the exponent x is any real number, is called an exponential function to the base b. EXPONENTIAL AND LOGARITHMIC FUNCTIONS y=2x, y=3x, y=(1/2)x x 2x 3x (½)x -2 ¼ 1/9 4 -1 ½ 1/3 2 0 1 1 1 1 2 3 ½ 2 4 9 ¼ 3 8 27 1/8 y=(1/2)x y=3x y=2x EXPONENTIAL AND LOGARITHMIC FUNCTIONS One of the most useful numbers that is used as a base in y=bx is a certain irrational number denoted by the letter e in honor of the Swiss mathematician and physicist Leonhard Euler (1707-1783). e is approximetely 2.71828. EXPONENTIAL AND LOGARITHMIC FUNCTIONS Example The predicted population P(t) of a city is given by P(t)=100,000e.05t where t is the number of years after 1980. Predict the population in the year 2000. t=20 and P(20)=100,000e.05(20)=100,000e1 P(20)=100,000e=271,828 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Example A mail order firm finds that proportion P of small towns in which exactly x persons respond to a magazine advertisement is given approximately by the formula .5 x e (.5) P 1.2.3..... x From what proportion of small towns can the firm expect exactly two people to respond? EXPONENTIAL AND LOGARITHMIC FUNCTIONS Example If we let P=g(x), then we want to find g(2) .5 e (.5) g ( 2) 1.2 2 e .5 (.5)2 (.60653)(.25) g ( 2) .07582 1.2 2 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Another important type of function is a logarithmic function, which is related to an exponential function. The logarithmic function base b, denoted logb is defined by y=logbx if and only if by=x The domain of logb is all positive numbers and its range is all real numbers. Logbx=y means by=x. Thus we can say that log28=is the logarithmic form of the exponential form 23=8. EXPONENTIAL AND LOGARITHMIC FUNCTIONS Logarithms to the base 10, called common logarithms, were frequently used for computational purposes before the pocket-calculator age. The subscript 10 is generally omitted from the notation. Thus logx means log10x Important in calculus are logarithms to the base e, called natural (or Naperian*) logarithms. We use the notation “ln” for such logarithms. Thus, lnx means logex ___________ * Scottish mathematician John Napier (1550-1617), the inventer of logarithms. EXPONENTIAL AND LOGARITHMIC FUNCTIONS Logarithmic scales reduce wide-ranging quantities to smaller scopes. For example, the decibel is a logarithmic unit quantifying sound pressure and voltage ratios. EXPONENTIAL AND LOGARITHMIC FUNCTIONS Properties of Logarithms 1. Logb(mn) = Logbm + Logbn m 2. log b log b m log b n n n log m n log b m 3. b 4. Logb1 = 0 5. Logbb = 1 EXPONENTIAL AND LOGARITHMIC FUNCTIONS Properties of Logarithms 6. log b b n n 7. If log b m log b n then m = n 8. If b b m n 9. b logb x x then m = n MATHEMATICS OF FINANCE PURPOSE: TO USE MATHEMATICS TO MODEL SELECTED TOPICS IN FINANCE THAT DEAL WITH TIME VALUE OF MONEY SUCH AS INVESTMENTS, LOANS, ETC. COMPOUND INTEREST WE SHALL USE MATHEMATICS TO MODEL SELECTED TOPICS IN FINANCE THAT DEAL WITH THE TIME-VALUE OF MONEY, SUCH AS INVESTMENTS, LOANS, ETC. PRACTICALLY EVERYONE IS FAMILIAR WITH COMPOUND INTEREST, WHEREBY THE INTEREST EARNED BY AN INVESTED SUM OF MONEY (OR PRINCIPAL-capital sum) IS REINVESTED SO THAT IT TOO EARNS INTEREST. THAT IS, THE INTEREST IS CONVERTED (OR COMPOUNDED) INTO PRINCIPAL AND HENCE THERE IS "INTEREST ON INTEREST”. COMPOUND INTEREST FOR EXAMPLE, SUPPOSE A PRINCIPAL OF TL 100 IS INVESTED FOR TWO YEARS AT THE RATE OF 5 PERCENT COMPOUNDED ANNUALLY. AFTER ONE YEAR THE SUM OF THE PRINCIPAL AND INTEREST IS 100 + .05(100) = TL 105. THIS IS THE AMOUNT ON WHICH INTEREST IS EARNED FOR THE SECOND YEAR, AND AT THE END OF THAT YEAR THE VALUE OF THE INVESTMENT IS 105 + .05(105) = TL 110.25. THE TL 110.25 REPRESENTS THE ORIGINAL PRINCIPAL PLUS ALL ACCRUED INTEREST; IT IS CALLED THE ACCUMULATED AMOUNT OR COMPOUND AMOUNT. COMPOUND INTEREST THE DIFFERENCE BETWEEN THE COMPOUND AMOUNT AND THE ORIGINAL PRINCIPAL IS CALLED THE COMPOUND INTEREST. IN THE ABOVE CASE THE COMPOUND INTEREST IS 110.25 - 100 = TL 10.25. COMPOUND INTEREST MORE GENERALLY, IF A PRINCIPAL OF P TL IS INVESTED AT A RATE OF 100r PERCENT COMPOUNDED ANNUALLY (FOR EXAMPLE, AT 5 PERCENT, r IS .05), THEN THE COMPOUND AMOUNT AFTER ONE YEAR IS P + Pr OR P(1 + r). AT THE END OF THE SECOND YEAR THE COMPOUND AMOUNT IS P(1 +r) + [P(1 +r)]r = P(1 + r)[1 + r] [FACTORING] = P(1 + r)2 COMPOUND INTEREST SIMILARLY, AFTER THREE YEARS THE COMPOUND AMOUNT IS P(1 + r)3. IN GENERAL, THE COMPOUND AMOUNT S OF A PRINCIPAL P AT THE END OF n YEARS AT THE RATE OF r COMPOUNDED ANNUALLY IS GIVEN BY S = P(1 + r)n COMPOUND INTEREST EXAMPLE 1 IF TL 1000 IS INVESTED AT 6 PERCENT COMPOUNDED ANNUALLY, A.FIND THE COMPOUND AMOUNT AFTER TEN YEARS. B.FIND THE COMPOUND INTEREST AFTER TEN YEARS. COMPOUND INTEREST EXAMPLE 1 FIND THE COMPOUND AMOUNT AFTER TEN YEARS. WE USE EQ. (S n = 10. = P(1 + r)n ) WITH P = 1000, r = .06, AND S = 1000(1 + .06)10 = 1000(1.06)10. WE FIND THAT (1.06)10 AS 1.790848. THUS, S ≈ 1000(1.790848) ≈ TL 1790.85. COMPOUND INTEREST EXAMPLE 1 FIND THE COMPOUND INTEREST AFTER TEN YEARS. USING THE RESULT FROM PART (A), WE HAVE COMPOUND INTEREST = S — P = 1790.85 - 1000 = TL 790.85. COMPOUND INTEREST EXAMPLE 2 SUPPOSE THE PRINCIPAL OF TL 1000 IN EXAMPLE 1 IS INVESTED FOR TEN YEARS AS BEFORE, BUT THIS TIME THE COMPOUNDING TAKES PLACE EVERY THREE MONTHS (THAT IS, QUARTERLY) AT THE RATE OF 1.5 PERCENT PER QUARTER. THEN THERE ARE FOUR INTEREST PERIODS OR CONVERSION PERIODS PER YEAR, AND IN TEN YEARS THERE ARE 10(4) = 40 CONVERSION PERIODS. THUS THE COMPOUND AMOUNT WITH R = .015 IS 1000(1.015)40≈ 1000(1.814018) ≈ TL 1814.02. COMPOUND INTEREST EXAMPLE 3 THE SUM OF TL 3000 IS PLACED IN A SAVINGS ACCOUNT. IF MONEY IS WORTH 6 PERCENT COMPOUNDED SEMIANNUALLY, WHAT IS THE BALANCE IN THE ACCOUNT AFTER SEVEN YEARS? (ASSUME NO OTHER DEPOSITS AND NO WITHDRAWALS.) COMPOUND INTEREST EXAMPLE 3 HERE P = 3000, THE NUMBER OF CONVERSION PERIODS IN 7(2) = 14, AND THE RATE PER CONVERSION PERIOD IS .06/2 = .03. BY EQ. (S = P(1 + r)n ) WE HAVE S = 3000(1.03)14 ≈ 3000(1.512590) ≈ TL 4537.77 COMPOUND INTEREST EXAMPLE 4 HOW LONG WILL IT TAKE FOR YTL 600 TO AMOUNT TO YTL 900 AT AN ANNUAL RATE OF 8 PERCENT COMPOUNDED QUARTERLY? COMPOUND INTEREST EXAMPLE 4 THE RATE PER CONVERSION PERIOD IS .08/4 = .02. LET N BE THE NUMBER OF CONVERSION PERIODS IT TAKES FOR A PRINCIPAL OF P = 600 TO AMOUNT TO S = 900. THEN FROM EQ. (S = P(1 + r)n ), 900 = 600(1.02)n, (1.02)n= 900/600 (1.02)n = 1.5. TAKING THE NATURAL LOGARITHMS OF BOTH SIDES, WE HAVE n ln (1.02) = ln 1.5, (Prop. logbmn=nlogbm) ln 1.5 .40547 n 20.478 ln( 1.02) .01980 THE NUMBER OF YEARS THAT CORRESPONDS TO 20.478 QUARTERLY CONVERSION PERIODS IS 20.478/4 = 5.1195, WHICH IS SLIGHTLY MORE THAN 5 YEARS AND 1 MONTH. COMPOUND INTEREST IF YTL 1 IS INVESTED AT A NOMINAL RATE OF 8 PERCENT COMPOUNDED QUARTERLY FOR ONE YEAR, THEN THE YTL WILL EARN MORE THAN 8 PERCENT THAT YEAR. THE COMPOUND INTEREST IS S - P = 1(1.02)4 – l ≈ 1.082432 - 1 = YTL .082432, WHICH IS ABOUT 8.24 PERCENT OF THE ORIGINAL YTL. THAT IS, 8.24 PERCENT IS THE RATE OF INTEREST COMPOUNDED ANNUALLY THAT IS ACTUALLY OBTAINED, AND IT IS CALLED THE EFFECTIVE RATE. COMPOUND INTEREST FOLLOWING THIS PROCEDURE, WE CAN SHOW THAT THE EFFECTIVE RATE r COMPOUNDED N TIMES A WHICH CORRESPONDS TO A NOMINAL RATE OF YEAR IS GIVEN BY n r Effective _ rate 1 1 n WE POINT OUT THAT EFFECTIVE RATES ARE USED TO COMPARE DIFFERENT INTEREST RATES, THAT IS, WHICH IS "BEST." COMPOUND INTEREST EXAMPLE 5 WHAT EFFECTIVE RATE CORRESPONDS TO A NOMINAL RATE OF 6 PERCENT COMPOUNDED SEMIANNUALLY? COMPOUND INTEREST EXAMPLE 5 THE EFFECTIVE RATE IS 2 .06 Effective _ rate 1 1 .0609 2 THE EFFECTIVE RATE IS 6.09 PERCENT. COMPOUND INTEREST EXAMPLE 6 TO WHAT AMOUNT WILL YTL 12,000 ACCUMULATE IN 15 YEARS IF INVESTED AT AN EFFECTIVE RATE OF 5 PERCENT? COMPOUND INTEREST EXAMPLE 6 SINCE AN EFFECTIVE RATE IS THE ACTUAL RATE COMPOUNDED ANNUALLY, WE HAVE S = 12,000(1.05)15 ≈ 12,000(2.078928) ≈ YTL 24,947.14. COMPOUND INTEREST EXAMPLE 7 HOW MANY YEARS WILL IT TAKE FOR A PRINCIPAL OF P TO DOUBLE AT THE EFFECTIVE RATE OF r ? COMPOUND INTEREST EXAMPLE 7 LET N BE THE NUMBER OF YEARS IT TAKES. WHEN P DOUBLES, THEN THE COMPOUND AMOUNT S IS 2P. THUS 2P = P(1 +R)N AND SO 2 = (1 + r)n, ln 2= n ln (1 + r). n ln 2 .69315 ln( 1 r ) ln( 1 r ) HENCE, .69315 .69315 n 11.9Years ln( 1.06) .05827 FOR EXAMPLE, IF R = .06, THEN THE NUMBER OF YEARS IT TAKES TO DOUBLE A PRINCIPAL IS APPROXIMATELY COMPOUND INTEREST EXAMPLE 8 SUPPOSE THAT YTL 500 AMOUNTED TO YTL 588.38 IN A SAVINGS ACCOUNT AFTER THREE YEARS. IF INTEREST WAS COMPOUNDED SEMIANNUALLY, FIND THE NOMINAL RATE OF INTEREST, COMPOUNDED SEMIANNUALLY, THAT WAS EARNED BY THE MONEY. COMPOUND INTEREST EXAMPLE 8 LET r BE THE SEMIANNUAL RATE. THERE ARE SIX CONVERSION PERIODS. THUS, 500(1 + r)6 = 588.38, (1 + r)6 = 588.38/500 6 r 588.38 1 .0275 500 THUS THE SEMIANNUAL RATE WAS 2.75 PERCENT, AND SO THE NOMINAL RATE WAS 5.5 PERCENT COMPOUNDED SEMIANNUALLY.