Honors PreCalculus
Chapter 4 – Exponential
and Logarithmic
Functions
Ms. Carey
Chapter 4 Assignments
11/1
11/5
11/12
11/19
TH:
p. 285 9-19o, 21-41eoo, 45, 53
F:
p. 297 11-18, 19, 21, 29, 31, 33, 43,45,49-55o, 61, 65
M:
p. 310 1-45o
T:
p. 310 53-60, 61, 63, 71, 81, 85-95o, 101, 103
W:
QUIZ 4.1-4.3
TH:
p. 321 29-39o (expand), 45-57eoo, 61-67o
F:
p. 327 1-11o, 17-33o
M:
W.S.
T:
4.4-4.5 QUIZ
W:
p. 335 3, 5, 7, 11-17o, 29-35o, 41
TH:
p. 347 1-9o, p. 355 1, 3, 9
F:
Review
M:
Review
T:
Ch 4 TEST
W:
NO SCHOOL
TH:
NO SCHOOL - Thanksgiving
F:
NO SCHOOL
Honors Pre-Calculus
4.1
One-to-One Functions & Inverse Functions
Learning Targets: Students will be able to identify one-to-one functions. Students will be able to graph, solve
analytically, and verify inverse functions.
Functions:
- To determine if a graph is a function, we use vertical line test
- To determine if a graph is a one-to-one function, it must also pass the horizontal line test.
Is this a one-to-one function?
Why or Why not?
12.
Characteristics of Inverse Functions
Domain of f(x) = Range of f 1 (x)
Range of f(x) = Domain of f 1 ( x)
Symmetric to the line y=x
16. Draw the inverse of the function
Draw the inverse of the function
Test to check if functions are inverse
Show
f
g  ( x) AND  g f  ( x)  x
Ex: Show that f ( x)  2 x and g ( x ) 
1
x are inverses.
2
30.
Show the following functions are inverses.
f ( x) 
x5
3x  5
and g ( x) 
2x  3
1 2x
How to find Inverse Functions
36.
Find the inverse f ( x)  x  1
38.
Find the inverse f ( x)  x  9 when x  0
3
2
Steps to finding inverse function
1. Rewrite function with y=
2. Switch x & y in the function
3. Solve for y
4. Watch the restrictions
Honors Pre-Calculus
4.2
Exponential Functions
Learning Targets: Students will be able to graph exponential functions and solve exponential functions using
a common base.
f ( x)  a x , where a > 0 and a  1
Exponential Function:
Graph on Calculator:
f ( x)  2 x
Graph by hand:
can this graph be negative?
can this graph be zero?
f ( x)  2 x  1
What happens if:
f ( x)  2 x  4
f ( x)  2 x 1
f ( x)  2 x  4
On the calculator graph:
f ( x )  3x 

f ( x)  4 x 
What happens? As the base gets larger, the graph gets steeper faster.
Special Exponential Function
f ( x)  e x
e is estimated at a value of 2.718… (more info on p. 293)
Law of Exponents
a a a
m
n
Properties of Exponential Functions
f ( x)  a x
mn
am
 a mn
n
a
domain is all real numbers
a 
range is positive real numbers
m n
a
m
 a mn
1 1
 m  
a
a
m
a0  1
if a m  a n , then m  n
no x-intercepts
y-intercept = 1
x-axis (y=0) is horizontal asymptote
GRAPH CONTAINS THE POINTS
1

(0,1) , (1, a ) and  1, 
a

Graph each function using transformations.
28.
f ( x)  1  3x  4
Ex:
y  2 x  1
Switching gears a little….
Solving Exponential Equations
To solve exponential equations, try to rewrite existing bases as powers of a common base, and then set the
exponents equal.
54.
92 x  27
Spreading of Rumors
44.
51 2 x 
1
5
56.
e 
4 x
e x  e12
2


.15 d
A model for the number of people N in a college community who have heard a certain rumor is N  P 1  e
, where
P is the total population of the community and d is the number of days that have elapsed since the rumor began. In a
community of 1000 students, how many students will have heard the rumor after 3 days?
Honors Pre-Calculus
4.3 – Day 1
Logarithmic Functions
Learning Targets: Students will be able to rewrite exponential functions to logarithmic functions and vice-versa.
Students will be able to solve logarithmic functions by rewriting them exponentially and be able to find its domain.
Logarithms are a way to convert problems so we may solve them.
a y  x converts to log a x  y
(read log base a of x equals y)
There are TWO bases that are used for logarithms
y  log a x
Common Log (Base 10)
 same as 10
y  log x
y
 x
answer
Natural Log
(Base e)
 same as e
y  ln x
y
 x
First you must be able to convert exponential to logarithmic and vice versa.
Convert to the other form of the equations
Examples:
2.
16  42
12.
e2.2  M
16.
logb 4  2
24.
ln x  4
Find the exact value WITHOUT the calculator
28.
log 3
1
9
base
argument
32.
log5 3 25
36.
ln e3
Find the domain of each function.
42.
g ( x)  log 1  x 2  1
2
46.
 x 
h( x)  log 3 

 x 1 
Arguments must be positive.
Honors Pre-Calculus
4.3 – Day 2
Logarithmic Functions
Learning Targets: Students will be able to graph logarithmic functions and solve logarithmic functions by
rewriting as an exponential function.
Exponential Form
Logarithmic Form
y  ax
y  log a x
Inverse Functions
Properties of Exponential Functions
Properties of Logarithmic Functions
Domain :
Domain:
Range:
Range:
x-intercepts:
x-intercepts:
y-intercept:
y-intercept:
asymptotes:
asymptotes:
Graph contains these points_________________________
Graph contains these points_________________
Graph both on this.
Graph using transformations.
f ( x)  ln( x  3)
62.
64.
f ( x)  ln( x)
82.
f ( x)  2  log( x  1)
Solve each Equation.
88.
log3 (3x  2)  2
92.
ln e 2 x  8
90.
log x
1
3
8
REMEMBER TO CHECK THAT
THE X VALUES DO NOT MAKE A
NEGATIVE ARGUMENT OR
NEGATIVE BASES!
Honors Pre-Calculus
4.4
Logarithmic Expansion, Condensing, and Change of Base
Learning Targets: Students will be able to expand and condense logarithmic functions. Students will be able
to solve logarithmic functions by using Change of Base.
Properties of Logarithmic Expansion
Exponential Equivalent
1.
log a 1  0 and ln1  0
0
0
- comes from the facts that a  1 and e  1
2.
log a a  1 and ln e  1
1
1
- comes from the facts that a  a and e  e
3.
a log a M  M and eln M  M
- comes from rewriting each as a logarithmic function
log a M  log a M and ln e M  ln M
log a a r  r and ln e r  r
- comes from rewriting logarithm in exponential form
5.
loga  MN   loga M  loga N
a r  a r and e r  e r (remember the understood e in the ln)
M
N
M N
- comes from the exponent rule a a  a
6.
M
log a 
N
aM
M N
- comes from the exponent rule N  a
a
7.
log a M r  r log a M
4.

  log a M  log a N

- proved by using x  log a M rewritten exponentially,
each side raised to the r power then simplified
8.
if loga M  loga N then M=N
M
N
- comes from the exponent rule if a  a then M=N
Examples:
Expand each logarithmic expression.
x
ex
32.
ln
38.
 3 x2  1 
log 5  2

 x 1 


36.

ln x 1  x 2

Write each expression as a single logarithm (CONDENSE)
50.
log  x 2  3x  2   2 log  x  1
58.
3log5  3x  1  2log5  2 x 1  log5 x
To help solve logarithmic expression, there is a formula called CHANGE OF BASE FORMULA.
Change of Base formula states that if you have a logarithmic expression, then you can rewrite it so you may evaluate it.
log a M 
log b M
log b a
log a M 
ln M
ln N
Notice that this formula allows you to
change the logarithm into any base. I
suggest that you use base 10 (common
base) or base e (natural log).
Use Change of Base formula to evaluate. (evaluate to 3 decimal places)
62.
log5 18
68.
log 2
On the N-Spire - Change of base is already installed. Ctrl-log will give you the opportunity to input the expression.
Honors Pre-Calculus
4.5
Logarithmic and Exponential Equations
Learning Targets: Students will be able to solve equations using logarithmic or exponential properties.
Remember log properties: if log a M  log a N , then M=N. Use logarithmic properties to solve the equations.
2.
log5  2 x  3  log5 3
Steps to Solve Logarithmically
6.
3log 2 x   log 2 27
32.
loga x  loga  x  2  loga  x  4
1. Get a single log expression
on each side.
2. If step one is not possible,
rewrite exponentially.
3. Solve for variable.
4. Check solution – Only
positive arguments and bases.
8.
2log3  x  4  log3 9  2
Ways to Solve Equations Exponentially
Two ways to solve equations exponentially
1.
Forcing a common base (previously used)
2.
Introducing natural log on each side
Steps to Solve by Introducing ln
18.
3x  14
1. Isolate the term with variable in it
2. Introduce ln on each side of
equation.
3. Using log properties, rewrite
equation so variable is out of
exponent.
4. Solve for variable
5. Check solution – Only positive
arguments and bases.
22.
2 x 1  51 2 x
28.
e x 3   x
30.
.3  42 x   .2
Honors Pre-Calculus
4.6
Compound Interest
Learning Targets: Students will be able to solve interest formulas using compound and continuous interest formulas.
Compound Interest Formula
The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year
is found by using the formula:
 r
A  P 1  
 n
nt
Continuous Compounding Interest Formula
The amount A after t years due to a principal P invested at an annual interest rate r is compounded continuously is
found by using the formula:
A  Pe rt
Find the amount that result in each investment:
6.
$700 invested at 6% compounded daily
after a period of 2 years.
Find the principal needed (present value).
14.
to get $800 after 3.5 years at 7% compounded monthly
8.
$40 invested at 7% compounded continuously
after a period of 3 years.
34.
How many years will it take for an initial investment of $25,000 to grow to $80,000? Assume a rate of interest
of 7% compounded continuously.
30.
How long does it take for an investment to double in value if it is invested at 10% per annum compounded
monthly?
… if compounded continuously?
40.
Tracy is contemplating the purchase of 100 shares of a stock selling for $15 per share. The stock pays no
dividends. Her broker says that the stock will be worth $20 per share in 2 years. What is the annual rate of return
on this investment?
Honors Pre-Calculus
4.7 & 4.8
Growth and Decay & Exponential, Logarithmic, & Logistic Models
Learning Targets: Students will be able to solve and apply interest formulas using compound and continuous interest
formulas.
4.7 Growth and Decay
Many natural phenomena have been found to follow the law that an amount A varies with time t according to the formula
A  Ao e kt where Ao stands for the original amount (when t=0) and k  0 is a constant specific to
each situation.
If k>0, then A is increasing over time, i.e. it will follow the law of uninhibited growth.
If k<0, then A is decreasing over time, i.e. it will follow the law of uninhibited decay.
Growth of Bacteria
2.
The number N of bacteria present in a culture at time t (in hours) obeys the function
N (t )  1000e.01t
a.
Determine the number of bacteria at t=0 hours.
b.
What is the growth rate of the bacteria?
d.
What is the population after 4 hours?
e.
When will the number of bacteria reach 1700?
f.
When will the number of bacteria double?
c.
Graph the function on a calculator.
Radioactive Decay
10.
The half-life of radioactive potassium is 1.3 billion years. If 10 grams is present now, how much will be present
in 100 years?
4.8 Exponential, Logarithmic, and Logistic Models
Data sometimes follows exponential, logarithmic or logistic models. Logistic Model is used to model situations where the
value of the dependent variable (y) is limited. An example of this is the populations of species are limited by the
availability of natural resources.
2.
A strain of E-Coli is placed into a petri dish at 30  C and allowed to grow. The following data are collected.
Theory states that the number of bacteria in the petri dish will initially grow according to the law of uninhibited growth.
The population is measured using an optical device in which the amount of light that passes through the petri dish is
measured.
a)
Draw a scatter diagram treating time as the predictor variable (on the graphing
Time (hrs), x Population, y
calculator)
2.5
0.175
3.5
0.38
b)
Using a graphing calculator, fit an exponential function to the data.
4.5
0.63
4.75
0.76
5.25
1.2
c)
Express the function found above in this form N (t )  N O e kt
d)
Graph the above function.
e)
Use the above function to predict the population at x = 6 hours.
f)
Use the above function to predict when the population will reach 2.1.
10.
The following data obtained from the U.S. Census Bureau represent the world population. An ecologist is
interested in finding a function that describes the world population.
Year
1993
1994
1995
1996
1997
1998
1999
2000
2001
Population
(in Billions)
5.531
5.611
5.691
5.769
5.847
5.925
6.003
6.08
6.157
a)
Use a graphing calculator to draw a scatter diagram.
b)
Using a graphing calculator, fit a logistic function to the data.
c)
Using the calculator to draw the function found in part b.
d)
Based on the fcn found in part (b), what is the carrying capacity of the world?
e)
Use the above fcn to predict the population of the world in 2010.
f)
When will the population be 7 billion?