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1 Exponential functions and logarithms 2 Overview A. 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 x y -4 2-4=1/24=1/16=0.0625 -3 2-3=1/23=1/8=0.125 -2 2-2=1/22=1/4=0.25 -1 2-1=1/2=0.5 0 20=1 0.25 0.5 0.75 20.25 = 21/4 = 4 2=1.1892… 20.5 = 21/2 = 2 =1.4142… 20.75 = 23/4 = 4 23 =1.6817… 1 21=2 2 22=4 3 23=8 4 24=16 4 Exponential function versus power function x is the exponent y=2x describes an exponential function x is the base y=x2 describes a (quadratic function), power function A power function is a function having an equation of the form y=xr (where r is a real number), i.e. x serves as the base. An exponential function is a function having an equation of the form y=bx (where b is a positive number distinct from 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, starting from the time of the investment). in the beginning: 1000 EUR each year: + 3% (of the preceding value) general formula??? t=1: A=1000+0.031000=1000+30=1030 t=2: A=1030+0.031030=1030+30.9=1060.9 t=3: A=1060.9+0.031060.9=1060.9+31.82…=1092.72… t=4: A=1092.72…+0.031092.72…=1092.72…+32.78…=1125.50… 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, starting from the time of the investment). t=1: A=1000+0.031000=1000+30=1030 A=1000+0.031000=1000(1+0.03)=10001.03=1030 t=2: A=1030+0.031030=1030+30.9=1060.9 A=1030+0.031030=1030(1+0.03)=10301.03 =10001.031.03=10001.032(=1060.9) t=3: A=1060.9+0.031060.9=1060.9+31.82…=1092.72… A=1060.9+0.031060.9=1060.9(1+0.03)=1060.91.03 =10001.031.031.03 =10001.033(=1092.72…) each year ×1.03 A=10001.03t 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, starting from the time of the investment). ‘each year: +3%’ corresponds to ‘each year ×1.03’ (1.03=1+3/100) we will use this formula also if t is not an integer A=10001.03t= 3 1000 1 100 multiple of an exponential function! t 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, starting from the time of the investment). yearly growth percentage=3% initial value=1000 A=10001.03 3 1000 1 100 t t= growth factor = 1.03 growth factor graph has J-form 10 Exponential growth cf. examples in parts A and B • A variable y grows exponentially iff y=y0bt (y0: initial value; b growth factor (b>0, b≠1)) • If y increases by p% every time unit (p: growth percentage), then cf. example in part B ♦ y grows exponentially ♦ growth factor is ♦ the equation is p b 1 100 p y y0 1 100 ♦ the graph has J-form t 11 Exercise: growth percentage/factor growth percentage (+ …% each time unit) growth factor (×… each time unit) +5% ×1.05 +50% ×1.5 +0.5% ×1.005 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, but since then its population decreased by 3% each year. Express the population N in terms of the time t (in years, starting from 1 Jan. 1950). t=1: N=1000-0.031000=1000(1-0.03)=10000.97=970 t=2: N=970-0.03970=970(1-0.03) =10000.970.97=10000.972 t=3: N=940.9-0.03940.9=940.9(1-0.03) =10000.973 N=10000.97t graph has reflected J-form 14 Exponential increase/decrease • If y decreases by p% every time unit (negative growth percentage), then ♦ y grows exponentially ♦ ♦ p growth factor is <1: b 1 100 t p the equation is y y0 1 100 ♦ the graph has reflected J-form • An exponential function y=bx is • increasing if b>1 • decreasing if b<1 cf. example 15 Exercise: growth percentage/growth (ctd.) growth percentage (+ …% each time unit) growth factor (×… each time unit) +5% ×1.05 +50% ×1.5 +0.5% ×1.005 –5% ×0.95 –50% ×0.5 –0.5% ×0.995 +100% ×2 +1000% ×11 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 such that … 10 x 1000 x 1000 10 990 10 x 1000 1000 x 100 10 10 x 1000 x in words: which exponent do you need to obtain 1000 when the base of the power is 10? log1000 3 3 is the (common) logarithm (or logarithm base 10) of 1000 18 Logarithms in words: log x is the exponent needed to make a power with base 10 equal to x (common) logarithm (logarithm base 10) of x: log x = y iff 10y = x Calculate the following logarithms (without calculator) log100 log1 000 000 log 0.001 log10 log1 log 100 log 0 10 100 ? ! 102 100 log100 2 log1 000 000 6 log 0.001 3 log10 1 log1 0 undefined undefined 19 Logarithms using the calculator Calculate the following logarithms and verify the result log 2 log 3 log 4 log 5 log 6 log8 log 9 log 20 log 3000 0.301 029 ... 0.477 121 ... 0.602 059 ... 0.698 970 ... 0.778 151 ... 0.903 089 ... 0.954 242 ... 1.301 029 ... 3.477 121 ... 20 Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms 1. Logarithm of a product 2. Logarithm of a quotient 3. Logarithm of a power F. Simple exponential equations G. More complicated exponential equations 21 Logarithm of a product log 2 log 3 log 4 log 5 log 6 log8 log 9 log 20 log 3000 0.301 029 ... 0.477 121 ... 0.602 059 ... 0.698 970 ... 0.778 151 ... 0.903 089 ... 0.954 242 ... 1.301 029 ... 3.477 121 ... log 6 log 2 log 3 log10 log 2 log 5 log 20 log 2 log10 log 3000 log1000 log 3 log 3000 3000 || 3 0.477... 10 1000 || ! 103 3 || 100.477... log1000 log 3 Logarithm of a product: loga b log a log b 22 Logarithm of a quotient log 2 log 3 log 4 log 5 log 6 log8 log 9 log 20 log 3000 0.301 029 ... 0.477 121 ... 0.602 059 ... 0.698 970 ... 0.778 151 ... 0.903 089 ... 0.954 242 ... 1.301 029 ... 3.477 121 ... log log636log log62log log33 log log10 log 5 log10 10 5 log 2 log 5 20 log 20 log 20 log10 10 log 2 log10 log 3000 log1000 log 3 log 3000 log 3 log1000 log1000 log 3000 log 3 log 3000 log 3000 log 3 3 log alog Logarithm of a product: quotient: log logbaa blog a blog b 23 Logarithm of a power log 2 log 3 log 4 log 5 log 6 log8 log 9 log 20 log 3000 0.301 029 ... 0.477 121 ... 0.602 059 ... 0.698 970 ... 0.778 151 ... 0.903 089 ... 0.954 242 ... 1.301 029 ... 3.477 121 ... log22 log 4 2 log 2 log23 log8 3 log 2 log32 log9 2 log3 ! log3 3 log 3 log 3 Logarithm of a power: log a r r log a 24 Overview A. Exponential functions B. Exponential growth C. Exponential decrease D. Logarithms E. Some rules for calculations with logarithms F. Simple exponential equations 1. Example: a growing capital 2. 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, starting from the time of the investment). When will the amount in the savings account be equal to 1500 EUR? A=10001.03t t? such that A=1500 exponential equation: unknown is in the exponent 1000 1.03t 1500 (divide by 1000) 1.03t 1.5 (take logarithm of both sides) t log(1.03 )log(1.5) t log1.03 log1.5 log1.5 t 13.7... log1.03 (apply log a r r log a ) Answer: After about 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: log = log ♦ apply rule for logarithm of a power: ∙ log = log ♦ divide both sides by log : = log log 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 1. Example: two growing capitals 2. 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=10001.03t J=9001.035t t? such that A=J 1000 1.03t 900 1.035t 1.03t 900 t 1.035 1000 t 900 1.03 1.035 1000 t 900 1.03 log log 1000 1.035 1.03 900 t log log 1.035 1000 900 log 1000 t 1.03 log 1.035 ( ) ( ) 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=10001.03t J=9001.035t 900 log 1000 t 1.03 log 1.035 t 21.7... Answer: It takes nearly 22 years before the two amounts are equal. 30 Solution procedure • To solve an exponential equation of the form 1 ∙ 1 = 2 ∙ 2 (where 1 , 2 , 1 and 2 are positive numbers) ♦ divide both sides by 1 : 1 ♦ divide both sides by 2 : 1 2 = 2 1 = ∙ 2 2 1 ♦ apply rule for quotient of powers: 1 2 = 2 1 ♦ finally, apply the procedure for simple exponential equations memorize the procedure, not the formulas!!!