Transcript Chapter 11 Power Point

*
* Objectives:
* Use the properties of exponents.
* Evaluate and simplify expressions containing
rational exponents.
* Solve equations containing rational exponents.
*
*
* A number is written in scientific notation when it is
in the form a x 10n where a is between 1 and 10 and
n is an integer.
*
At their closest points, Mars and Earth are approximately 7.5 x
107 kilometers apart.
a.
Write this distance in standard form.
75,000,000
b.
How many times further is the distance Mars Pathfinder
traveled than the minimum distance between Earth and
Mars? (the pathfinder traveled 4.013 x 108)
401,300,000
75,000,000
= .535 x 101 = 5.35
Please work with your table
partner to go through the
graphing calculator exploration.
**If we have time at the end**
*
*
*
a. 1251/3 =
b. (s2t3)5 =
c.
 2
 
5
1
5
e. (81c4)1/4 =
10 15
s t
 5
3

x
d.  y  
 x 4 3 


2
y
x9
f. 6253/4 =
3c
125
g. Simplify:
7
2
25 3
2 2
r s t
r 7s25t3 
r3s12t rst
h. Solve: 734 = x3/4 + 5
x  6561
Property
Product
Definition
nm
 a
n m
aa
Power of a Power
(a )  a
Power of a Quotient
am
am
( )  m
b
b
Power of a Product
n m
nm
(ab)  a b
m
m
m
Property
Quotient
Rational Exponent
Negative Exponent
Zero Exponent
Definition
m
a
m n

a
an
a
m/n
n
a

1
 n
a
a  1
0
n
a
m
WARM-UP:
*
* Learning Targets:
* I can solve problems involving exponential
growth and decay.
* I can evaluate expressions involving logarithms.
* I can solve equations and inequalities involving
logarithms.
*
*An exponential function is
of the form y = bx where the
base b is a positive real
number and the exponent is
a variable.
*
Please work with your table
partner to go through the
graphing calculator exploration.
*
*
b>1
Domain
Range
y- intercept
Behavior
Horizontal Asymptote
Vertical Asymptote
All real numbers
y  0
0<b<1
All real numbers
y  0
(0, 1)
(0, 1)
lim f(x)  
x 
Y=0
none
What happens when b=1?
lim f ( x )  0
x 
Y=0
none
*
y  log b x
b x
y
Write each expression in exponential form:
a.
b.
2
log 125 25 
3
1
log 8 2 
3
1
3
2
3
8 2
125  25
*
Write each equation in logarithmic form:
a.
b.
4  64
3
3
3
1

27
1
log 3
 3
27
log 4 64  3
*
* Evaluate the expression
1
log 7
y
49
1
7 
49
y
*
*
* How long would it take for 256,000 grams of
Thorium-234, with a half-life of 25 days, to decay to
1000 grams?
1 x
1000  256000( )
2
1
1 x
log(
)  log(( ) )
256
2
1000
1 x
( )
256000
2
1
1
log(
)  x log( )
256
2
1
1 x
( )
256
2
1
log(
)
256  x
1
log( )
2
x 8
200 years
*
Property
Definition
Product
log( mn)  log m  log n
Quotient
m
log( )  log m  log n
n
Power
log m  p log m
“Drop it down”
“lopEquality
it off”
p
If log m  log n, then m  n
*
Solve each equation (Warm-Up):
1
3
a.
1
log p 64 
2
b.
log 4 (2x  11)  log 4 (5x  4)
c.
log 11 x  log 11 ( x  1)  log 11 6
*
Solve each equation:
a.
1
3
1
log p 64 
2
You don’t HAVE to do anything fancy for
this one….just your brain…..ummmmmm
1
log p 4 
2
1
2
p  4 ????
p  16
p 4
*
Solve each equation:
b.
log 4 (2x  11)  log 4 (5x  4)
2x  11  5x  4
15  3x
5 x
*
Solve each equation:
c.
log 11 x  log 11 ( x  1)  log 11 6
log 11[ x( x  1)]  log 11 6
x( x  1)  6
2
( x  3)( x  2)  0
x  x6  0
x  3, 2
x x6
2
*
d.
log 2 x  log 2 ( x  1)  6
*
e.
log 7 x  6
f.
log 9 ( x  1)  6
9
3
d.
log 2 x  log 2 ( x  1)  6
log 2[ x( x  1)]  6
2 x x
6
2
2 x x
6
2
0  x  x  64
2
e.
log 7 x  6
9
7 x
6
9
117649  x
9
9
117649  x
3.659  x
f.
log 9 ( x  1)  6
3
log 11 x  log 11( x  1)  6
*
Section 11.2 & 4: Exponential and
Logarithmic Functions (Graphing)
Objectives:
Graph exponential functions and inequalities.
Graph logarithmic functions and inequalities.
* Graph the exponential functions y=4x, y=4x+2, and
y=4x-3 on the same set of axes. Compare and
contrast the graphs.
*
x
x
1
1
1
6

2
Graph the exponential functions y=   , y=  5  , and y=  5 
5
on the same set of axes. Compare and contrast the graphs.
*
x
* Graph y < 2x + 1.
*
*
y3
x
y  log 3 ( x  1)
*
y  log 5 x  2
*
*
* Solve: 32x-1 = 4x
* Solve 2log8 x – log8 3 = log8 10
y  log 2 ( x  4)
*
*
According to Newton’s Law of Cooling, the difference
between the temperature of an object and its surroundings
decreases in time exponentially.
Suppose a certain cup of coffee is 95oC and it is in a room
that is 20oC. The cooling for this particular cup can be
modeled by the equation y = 75(0.875)t where y is the
temperature difference (from the room) and t is the time in
minutes.
* Find the temperature of the coffee after 15 minutes.
y  75(.875)15
y  10.12o 20  10.12o  30.12o
The equation A =
,where A is the final amount,
P is the initial (principal) amount, r is the annual interest
rate, n is the number of times interest is paid/compounded
per year, and t is the number of time periods, is used for
modeling compound interest.
*
* Suppose that a researcher estimates that the initial
population of varroa (the parasite that kills bee
populations) in a colony is 500. They are increasing
at a rate of 14% per week. What is the expected
population in 22 weeks?
*
* Determine the amount of money in a money market
account providing an annual rate of 5% compounded
daily if Marcus invested $2000 and left it in the
account for 7 years.
*
*
* How long would it take for 256,000 grams of
Thorium-234, with a half-life of 25 days, to decay to
1000 grams?
1 x
1000  256000( )
2
1
1 x
log(
)  log(( ) )
256
2
1000
1 x
( )
256000
2
1
1
log(
)  x log( )
256
2
1
1 x
( )
256
2
1
log(
)
256  x
1
log( )
2
x 8
200 years
1.
Graph y = log4 (x+2) – 1
2.
Solve 103x – 1 = 6
3.
If I want to make $500 in 2 years, and I have
$2000 to invest now, what interest rate do I
need to convince my bank to give me if they
add interest semi-annually?
2x
* Objectives
* Find common logarithms of numbers
* Solve equations and inequalities using common
logarithms
* Solve real-world application with common
logarithmic functions.
*
* A common logarithm is a logarithm with a base of 10
*
*
* Given that log 7  0.8451
evaluate each logarithm.
a.log 7,000,000  log 7  log 1000000
 .8451 6
 6.85
b.b. log 0.0007
 3.15
*
Evaluate each expression
a.
log 5(2)3
 1.602
=
log 40
=
2 log 19  log 6  1.779
b.
19 2
log
6
* Graph y > log (x – 4).
10
*
* Graph y = log x + 2
* Graph y = log (x + 1) - 4
* Graph y = -log (x) + 5
* Graph y = 3log (x – 4)
*
If a, b, and n are positive numbers and neither a nor b is 1,
then the following equation is true:
log b n
log a n 
log b a
*
* Find the value of log 9 1043
using the change of base
formula
log 1043

log 9
 3.163
*
*
*
a.
Solve the following equations:
63x = 81
x  .8175
b.
54x = 73
x  .667
c.
x  .132
* WARM-UP
* Solve log
5 67
using change of base.
* Solve 51/ 4 x  3  94
* Objectives
* Use the exponential function y=ex
* Find natural logarithms of numbers
* Solve equations and inequalities using natural
logarithms
* Solve real-world applications with natural
logarithms.
*
The formula for exponential growth and decay (in terms
of e) is A  Pe rt : where A is the final amount, P is the
initial (principal) amount, r is the rate constant, and t is
time. (HINT: Look for the phrase “compounded
continuously” when using this formula.)
*
* On your calculator, find the e button.
What number
does the calculator give you?
2.718
The number e is JUST A NUMBER…just like
……….JUST A NUMBER.

* Compare the balance after 25 years of a $10,000
investment earning 6.75% interest compounded
continuously to the same investment compounded
semiannually.
y  10000e
(.0675)( 25)
 54059.48
.0675 ( 2 )( 25)
y  10000(1 
)
2
 52574.61
*
According to Newton, a beaker of liquid cools exponentially
when removed from a source of heat. Assume that the
initial temperature is 90oF and that r=0.275.
Write a function to model the rate at which the liquid cools.
 90e
(.275)( t )
*
*
b. Find the temperature of the liquid after 4 minutes
 90e
(.275)( 4 )
 270.37
c. Graph the function and use the graph to verify your
answer to part (b).
skip
* A natural logarithm is a logarithm with a base of e and is
denoted ln x.
*
Convert log 6 254 to a natural logarithm and evaluate.
ln 254
 3.09
ln 6
*
* Solve: 6.5 = -16.25 ln x
 .4  ln x
.4
e x
x  .67
*
*
Solve each equation or inequality by using natural logarithms.
a. 32x = 7x-1
b. 6 x 2 2  48
ln 32 x  ln 7 x 1
2 x ln 3  ( x  1) ln 7
(2 ln 3) x  x ln 7  ln 7
(2 ln 3  ln 7) x   ln 7
 ln 7
x
2 ln 3  ln 7
x  7.74
( x 2  2) ln 6  ln 48
x2  2 
ln 48
ln 6
x 2  4.16
x 2  4.16  0
 2.04  x  2.04