Transcript Chapter 5: Exponential and Logarithmic Functions

Daisy Song and Emily Shifflett
 5.1: Composite Functions
 5.2: One-to-One Functions; Inverse Functions
 5.3: Exponential Functions
 5.4: Logarithmic Functions
 5.5: Properties of Logarithms
 5.6: Logarithmic and Exponential Equations
 5.7: Compound Interest
 5.8: Exponential Growth and Decay Models; Newton’s Law; Logistic
Growth and Decay Models
 5.9: Building Exponential, Logarithmic, and Logistic Models from Data
 (f °g)(x)= f(g(x))
 Domain (f °g) is all numbers in the domain of
g(x) that coincide with domain of f(x).
 Function is one-to-one is x1 and x2 are two
different inputs of a function and x1 ≠ x2
 A function is one-to-one if, when graphed, a
horizontal line ever only hits one point.
 A function that is increasing on an interval I is
a one-to-one function on I.
 A function that is decreasing on an interval I
is a one-to-one function on I.
 Inverse function of f: correspondence of the range of
f back to the domain of f
 Domain f= range f-1, range f= domain f-1
 x1=y1, x2=y2
y1=x1, y2=x2 where y1≠y2 and x1≠x2
 Verifying the Inverse: f(f-1(x))= x and f-1(f(x))=x
 Finding the Inverse Function:
x=Ay+B
Ay+B=x
Ay=x-B
y= (x-B)/A
 f(x)=ax where a>0 and a≠1
 Laws of Exponents
(as)(at)= as+t
(as)t=ast
(ab)s=asbs
1s=1
a-s=1/as= (1/a)s
a0=1
 For f(x)=ax, a>0 and a≠1, if x is any real
number then f(x+1)/f(x)=a
 f(x)=ax+n
 n moves the function up and down the y-axis
 The number e: (1+(1/n))n=e
 y=ex-n
 n stretches the curve
 If ab=ac then b=c
 y=logax if and only if x=ay
 Domain: 0<x<∞ Range: -∞<x<∞
 y=lnx if and only if x=ey
 Graphing Logarithmic Functions:
 Plot y=ax of y=logax function
 Draw the line y=x
 Reflect graph
 Properties of Logarithmic Functions:




Domain is x>0, Range is all real numbers
x-intercept is 1. There is no y-intercept
The y-axis is a vertical asymptote
The function is decreasing if 0<a<1 and
increasing if a>1
 The graph contains the points (1,0), (a,1), and
(1/a, 1)
 The graph is smooth and continuous, with no
corners or gaps
 Graphing Inverse Logarithmic Functions:





Find domain of f(x) and graph
Find the range and vertical asymptote
Find f-1(x)
Use f-1 to find the range of f
Graph f-1
 loga1=0
 logaa=1
 alogaM=M
 logaar=r
 loga(MN)= logaM+logaN
 loga(M/N)= logaM-logaN
 logaMr= rlogaM
 logaM= logbM/logba (a≠1, b≠1)
 y=logax = x=ay if a>0, a≠1
 If logaM=logaN then M=N if M,N, and a are
positive and a≠1
 If au=av then u=v if a>0, a≠1
 Graphing calculator can be used to graph
both sides of the equation and find the
intersect
 Simple Interest Formula: I=Prt
 Interest=(amount borrowed)(interest
rate)(years)
 Payment periods





Annually: once per year
Semiannually: twice per year
Quarterly: four times per year
Monthly: 12 times per year
Daily: 365 times per year
 Compound Interest: Inew=Pold+Iold
 Compound Interest Forumula: A=P(I+(r/n))nt
 Amount= (Principal
(number
amount)(Interest+(rate/times
per
year))
per year)(years)
 Effective interest: annual simple rate of
interest= compounding after one year
 Present value: time value of money
 P=A(1+(r/n)-nt if compound
 P=Ae-rt if continuously compounded
 Law of Uninhibited Growth: A(t)=A0ekt A0=amount
when t=0 k≠0
 Amount(time)= (original amount)e(constant)(time)
 Uninhibited Radioactive Decay, and Uninhibited
Growth of Cells follow this model
 Half-life: time required for ½ of the substance to
decay
 Newton’s Law of Cooling: u(t)= T+(u0-T)ekt
 Temperature(time)=constant temp+ (original tempconstant temp)e(constant)(time)
 Logistic Model: P(t)=c/1+ae-bt
 Population(time)=carrying
capacity/1+(constant)e -constant(time)
 c=carrying capacity P(t) approaches c as t
approaches ∞
 Describes situations where growth and
decay is limited
 b>0
growth rate, b<0
decay rate
 Properties of Logistic Growth Function:







Domain is all real numbers
Range 0<y<c
No x-intercepts, y-intercept at P(0)
Two horizontal asymptotes at y=o and y=c
P(t) is increasing if b>0
P(t) is decreasing if b<0
Inflection point= point where the graph changes
direction (up to down for growth models and down
to up for decay models)= P(t) ½ c
 Smooth and continuous graph, no corners or gaps
 Same as finding line of best fit
 Draw a scatter diagram of the data to see the
general shape to determine what type of
function to use
 Many calculators have a Regression option
to fit an equation to data
 To fit: enter data into calculator. Use correct
regression to calculate model.