x y

-2 1/16

-1 1/4

0 1

1 4

2 16

x y

-2 5/4

-1 5/2

0 5

1 10

2 20

x y

-2 2/9

-1 2/3

0 2

1 6

2 18

x y

-2 9

-1 3

0 1

1 1/3

2 1/9

x y

-2 8

-1 4

0 2

1 1

2 1/2

x y

-2 16

-1 8

0 4

1 2

2 1

1. 2. 3. 4. 5. 6.

7. a) growth b) (0.3) 8. a) growth b) (0,4) 9. a) decay b) (0,5)

10. a) decay b) (0,.5) 11. a) growth b) (0,10) 12. a) decay b) (0,1)

13. a) decay b) (0,2) 14. a) growth b) (0,1/2) 15. a) growth b) (0,1)

16. $1,500(1+0.06)5 = $2,007.39 17. $300,000(1-0.2)5 = $98,304

18. $75,00(1+0.125)30 = $2,568,247.87 19. $29,000(1-0.12)7 = $11,851.59

20. $120,00(1+0.012)5 = $127374.80

21. $1,860,000(1-0.035)10 = $1,302,525.03

7.2 – Properties of Exponential Functions

&7.3 – Logarithmic Functions as

Inverses

7.2 – Properties of Exponential Functions

&7.3 – Logarithmic Functions as

Inverses

Natural base exponential functions are exponential functions with base e.

e ≈ 2.71828These functions are used in order to

describe continuous growth or decay.

Continuously Compounded Interest

A(t) = Pe(r)(t)

Interest Rate (annual)

Time in yearsPrincipa

l

Amount in account at

time t

Ex: Suppose your little brother won a contest at the start of 5th grade that deposited $3000 in an account that pays 5% annual interest compounded continuously. How much will he have when he starts high school four years later? (Express to the nearest dollar)

t =

r =

P =

A(t) =

Ex: Suppose you win $2000 in the lottery. If you deposit this money in an account that pays 5.1% interest, how much money will you have in 3 years?

t =

r =

P =

A(t) =

A logarithm base b of a positive number x satisfies the following definition:

For b > 0, b ≠ 1, logbx = y if and only if by = x

logb x can be read as “log base b of x”y is the exponent to which base b must be raised to

get x.

Ex: Write 100 = 102 in logarithmic form

Ex: Write 81 = 34 in logarithmic form

Ex: What is the value of log8 32 (without using a calculator)?

Ex: What is the value of log4 32 (without using a calculator)?

A common logarithm is a logarithm with base 10. So instead of writing log10 x, you can simply write log x

A logarithmic function is the inverse of an exponential function.

The graph showsy = 10x

and it’s inversey = log x

Note: (0, 1) and (1, 10) are on the graph of y = 10x

and (1, 0) and (10, 1) are on the graph of y = log x.

The function y = ex has an inverse as well. That inverse is called the

natural logarithmic function:

y = logex or y = ln x

Ex: Use the definition of a natural logarithm to write 54.598= e4 in natural logarithmic form

Ex: Use the definition of a natural logarithm to write 7.389 = e2 in natural logarithmic form

Homework

Worksheet questions 22 – 43