15 Days
One Day
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Exponential functions are those that have
constant base variablepower
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An example of an exponential function is
f ( x)  2 x
Domain : ( , )
Range : (0, )
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An Exponential Function f with base a
f ( x)  a x for all x in , where a  0 and a  1.
Domain : (, ) Range : (0, ) y - int : (0,1) H.A. : y  0.
Note : The exponentia l function is a 1 to 1 function.
Graph of f ( x)  a x for a  1
is an increasing function.
Graph of f ( x)  a x for 0  a  1
is a decreasing function.
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Sketch Graphs for the following function:
f ( x)  3 x
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Sketch Graphs for the following function:
1
f ( x)   
4
x
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We can shift exponential function using the
same patterns from before. Use locator point
x h
f
(
x
)

a
 k where (h, k  1) is the locator point.
(0,1).
y  k is the Horizontal Asymptote
f ( x)  4 x
f ( x)  4 x  2
f ( x)  4 x  2  3
f ( x)  a  x horizontal flip (across) y - axis
f ( x)  a x vertical flip (across) x - axis
- Both types of reflections will change the
position(s) of your intercepts and should be
done before shifting.
The exponentia l function f given by f ( x)  a x
for 0  a  1 or a  1 is one to one.
Thus the following conditions are satisfied for x1 and x2 :
1. If x1  x2 , then a x1  a x2 .
2. If a x1  a x2 , then x1  x2 .  We will use this one alot!
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To solve equations with variables in the
exponents we need to:
1. Re-write both sides as the same base
using exponent rules.
2. Set the exponents equal using condition 2
of our theorem on exponential functions.
3. Solve for the variable.
7 x6  73 x4
35 x 8  9 x  2
27 x 1  9 2 x 3
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Writing an exponential function f ( x)  ba given
the y-int and a point on the function.
x
1. Substitute the y-int into your equation and
solve for b.
2. Re-write your equation with a value for b.
3. Substitute other point into your equation
from step 2 and solve for a.
4. Re-write your equation with values for a
and b.
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Find an exponential function of the form f ( x)  ba
that has the given y-int and passes through
the point P
y  int 8; P(3,1)
x
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Read 4.1; pg292 (# 2,4,7,9,11b - h, 13,14,
16 - 18, 25 NO TI’s for the graphs)
Two Days
The Compound Interest Formula :
nt
 r
A  P1   where :
 n
A is the total amount after t years
P is the principal invested
r is the annual interest rate (as a decimal)
n is the number of interest periods per year
t is the number of years.
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You have an account that returns 7% annual
interest compounded monthly. If you invest
$1500 for a total of 10 years, how much
money will you have in the account?
If n is a positive integer, then
n
 1
1    e  2.71828 as n  .
 n
The Natural Exponentia l Function f is definded by
f ( x)  e x
for every real number x.
The natural exponentia l function is one of the most useful functions
in advanced mathematic s and applicatio ns.
The Continuous ly Compound Interest Formula :
A  Pe rt where :
A is the total amount after t years
P is the principal invested
r is the annual interest rate (as a decimal)
t is the number of years.
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An initial investment of $35000 is
continuously compounded at 8.5% interest.
How much is the investment worth after 5
years? After 15 years?
Law of Growth and Decay Formula :
Let q0 be the value of a quantity q at time t  0.
If q changes instantane ously at a rate proportion al
to its current va lue, then :
q  q(t)  q0 e rt
where r  0 is the rate of growth (or r  0 is the rate of decay)
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Since 1980, world population in millions closely fits
.0156x
the exponential function defined by y  4481e
where x is the number of years since 1980.
The world populations was about 5,320 million in
1990. How closely does the function approximate
this value?
Use this model to approximate the population in
2012.
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Read 4.2; pg 303 (# 1-7,9,11-13,15,19,21,25)
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pg 292 (# 32,37,38,(53 use TI));
pg 304 (# 22,24 (45 & 51 use TI))
Four Days
The following expressions are equivalent.
a x
y
log a x  y
Examples
1) 32=9
3) log 4 64  3
2) xa+b=9
4) log M G  r
log a x  y
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The expression above is read “The log of x
base a equals y”
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x>0 (the number you take a log of must be
positive)
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If you don’t see an “a” value the base is
assumed to be 10.
◦ log 4 = log10 4
A natural log (ln) has a base of e.
◦ ln 4=loge4
1)
2)
3)
log 5 125
10
ln e
log 2 ( x  3)  3
4)
log 2 (2x  7)  log 2 (3x 1)
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Pg 317 #1,3,9,11,14(skip e), 17-27 odd.
No TI
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Begin 4.3.2
log a a  n
n
Evaluate
1)
2)
log 2 2
e
3
ln 7
 1 
3) log 3 

 27 
a
loga n
n
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a)
b)
Remember that f(x)=logax and f(x)=ax are
inverse functions.
This means that logarithmic functions will
look like exponential functions except their
x’s and y’s will be flipped.
Graph f(x)=log2x
Graph f(x)=log2(x+2)-1
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Domain of f(x)=log x
Range of f(x)=log x
(0,∞)
(-∞,∞)
Remember you can’t take the log of a
negative number or zero.
However… logs can equal negative
numbers.
Ex: log(1/2)
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Pg 317 #4,10,12,16, 33(a-g) No calc
47,59,63,65
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Begin 4.3.3
◦ Idea: Rewrite in exponential form. Plug in y’s to
find x’s
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Graph f(x)=log4(2x-1)
Find asymptotes, intercepts, domain and
range
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Logarithmic Functions Worksheet
pg 59 # 8, 11 - 14
pg 60 1-3, 5 - 17, 20,21
graph 20 & 21(no TI)
One Days
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Loga(xy)= Logax+Logay
Loga(x/y)= Logax-Logay
Logaxn= nLogax
Note: log(x+y) is not equal to log (x)+ log(y)
Note: log(x-y) is not equal to log (x)-log(y)
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1)
Loga(xy)= Logax+Logay
Loga(x/y)= Logax-Logay
Logaxn= nLogax
Expand Each Log
Log3(4x)
Write as a single log
4) 3log(x)+2log(y)
2)
Ln(3e)
5)
½Ln(4x)-yLn(6)
3)
Log(2x3/y4)
6)
2ln(xy)-3ln(x)+6ln(y)
Solve
1) Log4(2x+4)= 2log43+ log45
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2)
ln(x)+ln(x+3)=½ln(324)
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Pg 328 #1-15 odd, 18, 20, 22-26
Three Days
3 5
x
log 10 ( x)
log a ( x) 
log 10 (a )
or
ln( x)
log a ( x) 
ln( a)
log 6 (64)
log 6 ( 4)
1
log( x  5)  log( 2 x  6)  log  
 x
3x2
3
7
2 x 1
e  4e  32  0
2x
x
e  4e
x
x
 3
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Read 4.5
pg 339 (# 1-3,5,9,10,17,18,20,41,42,45)
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pg 340 (# 11,13,15,21,22,25,31,32,43,44,57)
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How long does it take for an initial
investment of $5000 to grow to $60000 in an
account that earns 8.5% interest compounded
monthly?
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The populations N(t) (in millions) of the
United States t years after 1980 may be
approximated by the formula N (t )  227e.007t .
When will the populations be twice what is
was in 1980?
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A 100g sample of a radioactive substance has
a half life of 30 minutes. After how many
hours will 20g remain?
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Solving Equations Worksheet
Review for Quiz