1. VLEKHO-HONIM 1 Exponential functions and logarithms

2. VLEKHO-HONIM 2 A. Exponential functions and exponential growth

3. VLEKHO-HONIM 3 Example 1: the function y=2 x Table xy -42 -4 =1/16=0.0625 -32 -3 =1/8=0.125 -22 -2 =1/4=0.25 2 -1 =1/2=0.5 02 0 =1 0.252 0.25 =1.1892… 0.52 0.5 =1.4142… 0.752 0.75 =1.6817… 12 1 =2 22 2 =4 32 3 =8 42 4 =16 Graph

4. VLEKHO-HONIM 4 Exponential function versus power function y=2 x describes an exponential function A power function is a function having an equation of the form y=x r (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=b x (where b is a positive number distinct from 1), i.e. x is the exponent. x is the exponent x is the base y=x 2 describes a (quadratic function), power function

5. VLEKHO-HONIM 5 Example 2: 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+0.031000=1000+30=1030 t=2: A=1030+0.031030=1030+30.9=1060.9 t=3: t=4: t=5: A=1060.9+0.031060.9=1060.9+31.82…=1092.72… A=1092.72…+0.031092.72…=1092.72…+32.78…=1125.50… A=1125.50…+0.03 1125.50…=1125.50…+33.76…=1159.27… general formula??? in the beginning: 1000 EUR each year: + 3% (of the preceding value)

6. VLEKHO-HONIM 6 Example 2: 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+0.031000=1000+30=1030 t=2: A=1030+0.031030=1030+30.9=1060.9 t=3: A=1060.9+0.031060.9=1060.9+31.82…=1092.72… A=1000+0.031000=1000(1+0.03)=10001.03=1030 A=1030+0.031030=1030(1+0.03)=10301.03 =10001.031.03=10001.03 2 (=1060.9) A=1060.9+0.031060.9=1060.9(1+0.03)=1060.91.03 =10001.031.031.03 =10001.03 3 (=1092.72…) each year ×1.03 A=10001.03 t

7. VLEKHO-HONIM 7 Example 2: 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). A=10001.03 t = ‘each year: +3%’ corresponds to ‘each year ×1.03’ (1.03=1+3/100) multiple of an exponential function! we will use this formula also if t is not an integer graph has J-form

8. VLEKHO-HONIM 8 Example 2: 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). A=10001.03 t = growth factor initial value=1000 growth factor = 1.03 yearly growth percentage=3% graph has J-form

9. VLEKHO-HONIM 9 Exponential growth A variable y grows exponentially iff y=y 0 b t (y 0 : initial value; b growth factor (b>0, b≠1)) If y increases by p% every time unit (p: growth percentage), then ♦ y grows exponentially ♦ growth factor is ♦ the equation is ♦ the graph has J-form cf. examples 1 and 2 cf. example 2

10. VLEKHO-HONIM 10 Exercise growth percentage (+ …% each time unit) growth factor (×… each time unit) +5%×1.05 +50%×1.5 +0.5%×1.005

11. VLEKHO-HONIM 11 Example 3: decreasing population of a town A town had 100 000 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: t=2: t=3: N=1000-0.031000=1000(1-0.03)=10000.97=970 N=970-0.03970=970(1-0.03) =10000.970.97=10000.97 2 N=940.9-0.03940.9=940.9(1-0.03) =10000.97 3 A=10000.97 t graph has reflected J-form

12. VLEKHO-HONIM 12 Exponential increase/decrease If y decreases by p% every time unit (negative growth percentage), then ♦ y grows exponentially ♦ growth factor is <1: ♦ the equation is ♦ the graph has reflected J-form An exponential function y=b x is increasing if b>1 decreasing if b<1 cf. example 3

13. VLEKHO-HONIM 13 Exercise 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

14. VLEKHO-HONIM 14 A. Exponential functions and exponential growth Handbook Chapter 4: Exponential and logarithmic functions 4.1 Exponential functions introduction and definition examples 1, 2, 3, 6 and 7 problems 16, 18, 19, 20, 30, 31, 32, 33, 34, 35, 36

15. VLEKHO-HONIM 15 B. Logarithms

16. VLEKHO-HONIM 16 Example Find x such that … 3 is the (common) logarithm (or logarithm base 10) of 1000 in words: which exponent do you need to obtain 1000 when the base of the power is 10?

17. VLEKHO-HONIM 17 Logarithms (common) logarithm (logarithm base 10) of x: log x = y iff 10 y = x in words: log x is the exponent needed to make a power with base 10 equal to x Calculate the following logarithms (without calculator) undefined

18. VLEKHO-HONIM 18 Logarithms using the calculator Calculate the following logarithms and verify the result

19. VLEKHO-HONIM 19 Some rules for calculations with logarithms Logarithm of a product:

20. VLEKHO-HONIM 20 Some rules for calculations with logarithms Logarithm of a power:

21. VLEKHO-HONIM 21 C. Exponential equations

22. VLEKHO-HONIM 22 Example 1: 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=10001.03 t t? such that A=1500 log( ) (apply ) exponential equation: unknown is in the exponent Answer: After about 13.7… years, the amount is equal to 1500 EUR. (divide by 1000) (take logarithm of both sides)

23. VLEKHO-HONIM 23 Example 2: two growing capitals log ( ) A=10001.03 t t? such that A=J J=9001.035 t 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?

24. VLEKHO-HONIM 24 Example 2: two growing capitals Answer: It takes nearly 22 years before the two amounts are equal. A=10001.03 t J=9001.035 t 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?