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Exponential functions and logarithms

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  1. Exponential functions and logarithms

  2. Overview • Exponential functions 1. Example: the function y=2x 2. Exponential function versus power function B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations G. More complicated exponential equations

  3. Example: the function y=2x Table Graph

  4. Exponential function versus power function x is the exponent x is the base y=2xdescribesanexponentialfunction y=x2describes a (quadraticfunction), power function A power functionis a functionhavinganequation of the form y=xr (where r is a realnumber), i.e. x serves as the base. An exponentialfunctionis a functionhavinganequation of the form y=bx (where b is a positivenumberdistinctfrom 1), i.e. x is the exponent.

  5. Overview A. Exponential functions B. Exponential growth 1. Example: a growing capital 2. Exponential growth 3. Exercise: growth percentage and growth factor C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations G. More complicated exponential equations

  6. Example: a growing capital An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, startingfrom the time of the investment). in the beginning: 1000 EUR eachyear: + 3% general formula??? (of the precedingvalue) t=1: A=1000+0.031000=1000+30=1030 A=1030+0.031030=1030+30.9=1060.9 t=2: A=1060.9+0.031060.9=1060.9+31.82…=1092.72… t=3: A=1092.72…+0.031092.72…=1092.72…+32.78…=1125.50… t=4: t=5: A=1125.50…+0.03 1125.50…=1125.50…+33.76…=1159.27…

  7. Example: a growing capital An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, startingfrom the time of the investment). t=1: A=1000+0.031000=1000+30=1030 A=1000+0.031000=1000(1+0.03)=10001.03=1030 A=1030+0.031030=1030+30.9=1060.9 t=2: A=1030+0.031030=1030(1+0.03)=10301.03 =10001.031.03=10001.032(=1060.9) A=1060.9+0.031060.9=1060.9+31.82…=1092.72… t=3: A=1060.9+0.031060.9=1060.9(1+0.03)=1060.91.03 =10001.031.031.03 =10001.033(=1092.72…) A=10001.03t each year ×1.03

  8. Example: a growing capital An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, startingfrom the time of the investment). ‘eachyear: +3%’ corresponds to ‘eachyear ×1.03’ (1.03=1+3/100) we willusethisformulaalsoif t is notan integer A=10001.03t= multiple of anexponentialfunction! graph has J-form

  9. Example: a growing capital An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, startingfrom the time of the investment). yearlygrowth percentage=3% initialvalue=1000 A=10001.03 t= growth factor growth factor = 1.03 graph has J-form

  10. Exponential growth cf. examples in parts A and B • A variable y growsexponentiallyiff y=y0bt • (y0: initialvalue; b growth factor (b>0, b≠1)) • If y increasesby p% every time unit • (p: growth percentage), then • y growsexponentially • growth factor is • the equation is • the graph has J-form cf. example in part B

  11. Exercise: growth percentage/factor

  12. Overview A. Exponential functions B. Exponential growth C. Exponential decrease 1. Example: decreasing population 2. Exponential increase/decrease 3. Exercise: growth percentage/factor (continued) D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations G. More complicated exponential equations

  13. Example: decreasing population A small village had 1000 inhabitants on 1 Jan. 1950, butsincethenitspopulationdecreasedby 3% eachyear. Express the population N in terms of the time t (in years, startingfrom 1 Jan. 1950). t=1: N=1000-0.031000=1000(1-0.03)=10000.97=970 N=970-0.03970=970(1-0.03) t=2: graph has reflectedJ-form =10000.970.97=10000.972 t=3: N=940.9-0.03940.9=940.9(1-0.03) =10000.973 N=10000.97t

  14. Exponential increase/decrease cf. example • If y decreasesby p% every time unit • (negativegrowth percentage), then • y growsexponentially • growth factor is <1: • the equation is • the graph has reflectedJ-form • An exponentialfunction y=bx is • increasingif b>1 • decreasingif b<1

  15. Exercise: growth percentage/growth (ctd.)

  16. Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms 1. Example 2. Logarithms 3. Logarithms using the calculator E. Some rules for calculations with logarithms F. Simple exponential equations G. More complicated exponential equations

  17. Example Find x suchthat … in words: which exponent do you need to obtain 1000 when the base of the power is 10? 3 is the (common)logarithm (or logarithm base 10) of 1000

  18. in words: log x is the exponent needed to make a power with base 10 equal to x Logarithms (common) logarithm (logarithm base 10) of x: log x = y iff 10y = x Calculate the followinglogarithms (without calculator) undefined undefined

  19. Logarithms using the calculator Calculate the followinglogarithms and verify the result

  20. Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms • Logarithm of a product • Logarithm of a quotient • Logarithm of a power F. Simple exponential equations G. More complicated exponential equations

  21. Logarithm of a product Logarithm of a product:

  22. Logarithm of a quotient Logarithm of a product: Logarithm of a quotient:

  23. Logarithm of a power Logarithm of a power:

  24. Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations • Example: a growing capital • Solution procedure G. More complicated exponential equations

  25. Example: a growing capital An amount of 1000 EUR is invested in a savings account yielding 3% of compound interest each year. Express the amount A in the savings account in terms of the time t (in years, startingfrom the time of the investment). Whenwill the amount in the savings account beequal to 1500 EUR? A=10001.03t t? suchthat A=1500 exponential equation: unknown is in the exponent (divideby 1000) (takelogarithm of bothsides) log( ) log( ) (apply ) Answer: Afterabout 13.7… years, the amount is equal to 1500 EUR.

  26. Solution procedure • An exponential equation is an equation in which the unknown appears at least once in an exponent. • To solve an exponential equation of the form (where a, b and c are positive numbers) • divide both sides by a: • take the logarithm of both sides: • apply rule for logarithm of a power: • divide both sides by memorize procedure, not formulas!!!

  27. Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations G. More complicated exponential equations • Example: two growing capitals • Solution procedure

  28. Example: two growing capitals Ann invests an amount of 1000 EUR in a savings account yielding 3% of compound interest each year. John invests 900 EUR in a savings account yielding 3.5% of compound interest each year. When will they have the same amount in their savings account? A=10001.03t J=9001.035t log() log() t? suchthat A=J

  29. Example: two growing capitals Ann invests an amount of 1000 EUR in a savings account yielding 3% of compound interest each year. John invests 900 EUR in a savings account yielding 3.5% of compound interest each year. When will they have the same amount in their savings account? A=10001.03t J=9001.035t Answer: Ittakesnearly 22 yearsbefore the twoamounts are equal.

  30. Solution procedure • To solve an exponential equation of the form (where , , and are positive numbers) • divide both sides by : • divide both sides by : • apply rule for quotient of powers: • finally, apply the procedure for simple exponential equations memorize the procedure, not the formulas!!!