Transcript 04-Powers_Roots_and_Logarithms-

POWERS, ROOTS AND LOGARITHMS
What is the relationship between powers, roots and logarithms?
INVERSES FUNCTIONS
AND PROPERTIES OF EXPONENTS
How do you find an inverse of an equation?
UNIT ACTIVATION: FINDING INVERSES
Find the inverse of
y = 2x + 3
x = 2y + 3
x – 3 = 2y
(x-3) = y-1
2
1)
2)
Switch the x and y
Solve for the new y
also known as f-1(x)
PROPERTIES OF EXPONENTS (
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)
x0 = 1  x≠ 0
x-1 =
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xn ∙xm = xn+m
= xn-m
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(xn)m = xnm
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TYPED ON YOUR PAPER
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when taking the root of a variable
you can’t take the even root of a neg #
EXAMPLES
x2 = 49
EXAMPLES
EXAMPLES
EXAMPLES
rationalize
solve
HOMEWORK
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Worksheet 1
LOGARITHMS AND THEIR PROPERTIES
What is the relationship between logarithms and exponential functions?
ACTIVATION: FINDING INVERSES OF THE
EXPONENTIAL EQUATION
What do we do with problems like
the last one on the homework
y = 3x
x = 3y
log x = log 3y
Not as easy to solve for y when y is the exponent so we remember the
primary rule of equations: whatever we do to one side we must do to the
other. In this case we take the logarithm of both sides
PRIMARY RULE FOR LOGARITHMS
logb x = y
becomes x = by
Solve:
log2 4 = x
log2 x3 = 3
log 1000 = x
EXAMPLES
log16 64 = x
log4
=x
log .1 = x
HOMEWORK
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Worksheet 2
LAWS OF LOGARITHMS
How are the laws of logarithms related to the properties of exponents?
ACTIVATION: INITIAL RULES FOR LOGARITHMS
Primary rule of logs:
logb x = y
becomes
x = by
What would be true of the following and WHY????
loga x = 0
loga a = 1
NOTE: Can’t take the log of a negative number
i.e. in logb x = y the x can’t be negative why?
HOW ARE THE LAWS FOR EXPONENTS AND
LOGARITHMS RELATED
let
convert
multiply
b = logax and c = logay
x=ab
y = ac
xy =abac
xy = ab+c
loga xy =loga ab+c
convert
loga xy = b + c
substitute
loga xy = logax + loga y
BY THE SAME TYPE OF PROOFS
x
log a  log a x - log a y
y
log a x n  n log a x
EXAMPLES
log (x2 -1) – log (x+2) = 1
log (4x -4) =2
log x
HOMEWORK
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Worksheet 3
ACTIVATION:
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What is the difference between these three
problems and how does that impact the way
you work with them?
log 50 + log 2
log x = log 12 – log 3
log 8 – log x = 2
HOMEWORK
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Worksheet 4
SOLVING EXPONENTIAL EQUATIONS
AND THE CHANGE OF BASE THEOREM
How can logarithms assist in solving an exponential equation?
ACTIVATION:
What is THE primary rule of equations
—whatever you do to one side you must do to the
other.
3=4x
log 3=log 4x
3=4x
ln 3=ln 4x
EXAMPLES:
After using the circular method, you see are you
back to solving exponential equations.
log4 3=x
3=4x
log 3=log 4x
log 3=x log 4
log 3=x
log 4
HOMEWORK
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Worksheet 5
REAL WORLD APPLICATIONS OF EXPONENTIAL
AND LOGARITHMIC EQUATIONS?
What are the real world applications of exponential and logarithmic
equations?
ACTIVATION:
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r
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A  P 1  
n
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nt
I = Prt
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A= final amount
P = principal
r = rate as a decimal
decimal
n = number of times compounded in one year
t = the time in years
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How are they the same and how are they different:
I = interest
P = principal
r = rate as a
t = time in years
EXAMPLES:
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In 1900, the population of the U.S. was 3,465,000 with an
annual growth rate of 6.2%. How long will it be until the
population reaches 10,000,000?
EXAMPLES:
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A certain bacteria colony has a growth rate of 26% per hour. If
there were 42 bacteria in the colony when the study began,
how long will it take to have 258 bacteria?
EXAMPLES:
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In 2000, the population of a county in Southeastern PA was
5,263,126. The population of this area has been decreasing at
a rate of 3% per year, if this continues, when will the population
go below 4,500,000?
HOMEWORK
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Worksheet 6
REVIEW
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Worksheet 7