discrete exponential growth

represented by points

Ex: most species only reproduce once a year

continuous exponential growth

represented by a line or curve etc.

Ex: over-lapping generations. some species bred thruout the year

exponential function

y = ab^{x}

y - amount after x period

a - intial value

b - growth factor

x - time period

how to write exponential functions and predict

2.9 mill ppl in 1980, increasing 1.7% each yr

a. write an exponential function

b. when will pop reach 4.5 mill?

a.

- find growth factor b

pop @ end of yr = 100% of pop @ start + 1.7% of pop @ start

pop @ end of yr = 101.7% pop @ start of year

b = 1.017

- y = ab
^{x}

y = (2.9)(1.017)^{x}

b.

- Use a graph
- Graph the equation y = (2.9)(1.017)
^{x} - x-value of 0 = 1980, x-value of 26 = 2006

- Graph the equation y = (2.9)(1.017)
- Use recursion
- Enter initial amount 2.9 & repeatedly multiply by growth factor (1.017)
- Count # times pressed Enter (260
- 1980 + 26 = 2006

exponential growth

y = ab^{x}

b > 1

exponential decay

y = ab^{x}

0 < b < 1

negative exponents

b^{-x} = (1/b)^{x}

Ex: y = 52(6/5)^{-x} = 52(5/6)^{x}

half-life questions

1200 ppl; every 1/2 hour, 1/2 ppl leave

1 half-life → (1200)(1/2)

2nd half-life → (1200)(1/2)(1/2)

f(x) = 1200(1/2)^{x}

how many ppl after 3 hrs?

about 18

fractional exponent rules

a^{1/n} = n[a Ex: 4^{1/3} = ^{3}[4

a^{m/n} = (^{n}[a)^{m} or ^{n}[a^{m} Ex: 5^{3/4} = (^{4}[5)^{3} or ^{4}[5^{3}

IF N IS EVEN, MUST USE ABSOLUTE VALUES

solving exponent algebraic equations

Find the value of b when f(x) = 4b^{x} and f(3/4) = 32

4b^{3/4} = 32

b^{3/4} = 8

^{3/4(4/3) }= 8

^{4/3}

^{4/3}= (

^{3}[9)

^{4}= 2

^{4}= 16

e

the number e is an irrational # that = approx. 2.718281828 1, (1+1/2)^{2}, (1+1/3)^{3}, ...

logistic growth function

a function in which the rate of growth of a quantity slows down after initially increasing/decreasing exponentially

solving e problems

spread of flue in 1,000 ppl modeled by y = 1000/(1 + 990e^{-0.7x}), where y = the # of ppl after x days

a. how many ppl after 9 days?

Graph the equation

Read y when x = 9 -> abt 355ppl

b. horizontal asymptotes?

Trace along same graph

min. y-value gets close to but never reaches 0

max. y-value gets close to but never reaches 1000

y = 0, y = 1000

c. Estimate max # of ppl

max. of y = 1000 ppl

inverse functions

two functions f & g r inverse functions if g(b) = a whenever f(a) = b

graph = reflection of the graph of the original function over the line y = x

inverse of f(x) can be written f^{-1}(x) or f^{-1} (although the exp. -1 usually means reciprocal, f^{-1}(x) is NOT 1/f(x)

graphing inverse functions

**a. **y = 2^{x }**b.** y = x^{2}

1. Plot pts to graph each function. Then interchange the coordinates of the original points and plot these points.

**a. **(1,2) → (2,1)

(-1, 1/2) → (1/2, -1)

**b. **(2,4) → (4,2)

(-2,4) → (4,-2)

deciding existence of inverse functions

an inverse is a function. if a reflection of a function over the line y = x isn't a function, then the inverse doesn't exist

**a. **y = 2^{x} **b. **y = x^{2}

1. Use the vertical line test to decide whether a reflection is the graph of a function (*inverse*)

**-or-**

2. Just like vertical line tests, u can use the horizontal line test to see whether a function has an inverse (*original*)

writing inverse equations

Ex: Write an equation for the inverse function of f(x) = 2/3x - 4

- f(x) = 2/3x - 4
- y = 2/3x - 4
- x = 2/3y - 4
- x + 4 = 2/3y
- (3/2)x + (3/2)4 = (3/2)(2/3)y
- 3/2x + 6 = y
- f
^{-1}x = 3/2x + 6

so just interchange x & y, and solve for y

logarithmic function

the inverse of the exponential function f(x) = b^{x} and written as f^{-1}(x) = log_{b}x

x = b^{a} → a = log_{b}x

x on outside, b in middle, a in center

base of a log can be any pos. # **except** 1

log_{4}(-2) is **undefined!** can't have neg log

logarithm

(of any pos. real # x) the exponent a when u write x as a power of a base b

x = b^{a} iff a = log_{b}x

base of a log can be any pos. # **except** 1

common logarithm

a log w/ base 10

common log of x written as log x

x = 10^{a} → a = log x

natural logarithm

a log w/ base e

usually written as ln x

x = e^{a} → a = ln x

evaluating logs

Evaluate each. (*Find a*). Round decimal answers to the nearest hundredth

a. log_{4}64

log_{4}64 = log_{4}4^{3} → a = 3; has to be 4^{3} cuz base = 4

log_{4}64 → 4^{a} = 64 → a = 3

b. log(1/10,000)

log(1/10000) = log10^{-4} → a = -4

c. ln5.3

use calc; = abt 1.67

d. log145

use calc; = abt 2.16

solving logs

**when solving these, remember that x isn't necessarily the x in the formula. it can be a, b, or x**

solve log_{x}81 = 2

log_{b}81 = 2

b^{2} = 81

b = 9

Solve. round decimal answers to nearest hundredth

a. 10^{x} = 15

10^{a} = 15

a = log 15 = abt 1.18

b. e^{a} = 29

e^{a} = 29

a = ln 29 = abt 3.37

c. log_{2}x + log_{2}(x-2) = 3

log_{2}x + log_{2}(x-2) = 3

log_{2}x(x-2) = 3

x(x-2) = 2^{3}

x^{2} - 2x = 8

x^{2} - 2x - 8 = 0

(x+2)(x-4) = 0

x + 2 = 0 or x - 4 = 0

x = -2 x = 4

**check for extraneous solutions!** Sub possible solutions into the original equation to be sure they're not extraneous

x = -2 is undefined, cuz no neg logs log_{2}(-2)

properties of logs

M, N, and P are pos. #'s w/ b not equaling 1, and k is any real #

Product of Logarithms Property

log_{b}MN = log_{b}M + log_{b}N

Quotient of Logarithms Property

log_{b}M/N = log_{b}M - log_{b}N

Power of Logarithms Property

log_{b}M^{k} = k log_{b }M

writing in terms of logM and logN

a. logM^{2}N^{3}

logM^{2} + logN^{3} = 2logM + 3logN

b. log(^{3}[M/N^{4})

log^{3}[M - logN^{4} = logM^{1/3} - logN^{4} = 1/3logM - 4logN

simplifying logs

ln18 - 2ln3 + ln4

ln18 - ln3^{2} + ln4 → power prop.

ln18 - ln9 + ln4

ln18/9 + ln4 → quotient prop.

ln2 + ln4

ln(2•4) → product prop.

ln8

log and exp word problems

###
- L = 10log(I/I
_{0}) What is the change in loudness(L) when I is doubled?

_{0}) What is the change in loudness(L) when I is doubled?L_{1 }= original loudness L_{2} = loudness after intensity is doubled

increase in loudness = L_{2} - L_{1}

L_{2} - L_{1} = 10log(2I/I_{0}) - 10log(I/I_{0})

10log(2•I/I_{0}) - 10log(I/I_{0})

10(log2 + logI/I_{0}) - 10log(I/I_{0})

10log2 + 10log(I/I_{0}) - 10log(I/I_{0})

10log2 = abt 3

- A(t) = A
_{0}(0.883)^{t }with A(t) - amount present t thousand yrs after death, A_{0}- amount @ time of death, and t - amount time since death (in 1000's of yrs)

found in 1968, died 8000 yrs ago, 37% present now; estimate age of the bone

A_{0}(0.883)^{t} = A(t)

A_{0}(0.883)^{t }= 0.37A_{0}

0.883^{t} = 0.37

log(0.883)^{t} = log0.37

t log 0.883 = log0.37

t = log0.37/log0.883

t = 8