Logandexponentspart2.notebook
December 16, 2013
Day 1
y = a x is the inverse of y = log a x
A.) Basic Log equations
1.) log 2 x = 6
2.) log x 49 = 2
3.) log 2 32
4.) 7 - log 3 243
Feb 13­12:12 PM
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5.) log6 1 = x
36
6.) log ¼ x = - 3
7.) log 1/8 x = - 1/3
8.) log x = -3
Feb 13­12:13 PM
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try these on your own...
9.) log x 64 = 3
10.) log 4 x = 1.5
Feb 13­12:15 PM
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December 16, 2013
Day 2
http://en.wikipedia.org/wiki/E_(mathematical_constant)#History
http://mathforum.org/dr.math/faq/faq.e.html
What is e? Who first used e? How do you find it? How many digits does it have?
e = 2.71828..., the Base of Natural Logarithms e is a real number constant that appears in some kinds of mathematics problems. Examples of such problems are those involving growth or decay (including compound interest), the statistical "bell curve," the shape of a hanging cable (or the Gateway Arch in St. Louis), some problems of probability, some counting problems, and even the study of the
distribution of prime numbers. It appears in Stirling's Formula for approximating factorials. It also shows up in calculus quite often, wherever you are dealing with either logarithmic or exponential functions. There is also a connection between
e and complex numbers, via Euler's Equation </dr.math/faq/faq.euler.equation.html>
. Let's look at a chart...
n
( 1 + 1/n)n
10
100
1,000
10,000
100,000
Dec 16­7:53 AM
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Evaluate
1. e0.08
2.) e -0.08
3.) e 2/3
4.) e 3.2
5.) e√2
Dec 16­9:15 AM
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Why study e?
Loge = ln
Find the answer to 3 significant figures
1.) ln 4
try it as
Loge 4
2.) ln
try it as
Loge
Dec 16­9:17 AM
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Logandexponentspart2.notebook
December 16, 2013
Day 3
B.) Special Case
What do we do when the base is the number e?
***
ln ex = x
eln x = x
examples
1.) e n = 75
type ln 75
2.) e n = 102
3.) ln x = 17
type e17
4.) ln e4
5.) log x = 2.7
Feb 13­12:16 PM
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C.) Change of Base Formula
easy...
Formula
1.) log 2 32
Log b a =___________
challenging...
Find the answer to 3 significant figures
2.) log 4 77
3.) log 5 75
Jan 13­9:30 AM
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Logandexponentspart2.notebook
December 16, 2013
Day 4
Product Property
Log b uv = log b u + logb v
Quotient Property
Log b
= log b u - logb v
Power Property
Log b u r = r log b u
Given on IB exam 2014!!!!!
Feb 13­2:31 PM
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A.) Examples
Expand each
1.) log m2n
2.) log m2
n
3.) log
4.) log
try
5.) log
Feb 13­2:33 PM
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B.) Condense
6.) log 12 - log 3
8.) Log 3 5 + 9
try...
10.) log3 45 -log3 15
7) Ln 3 + 2 Ln 3
hint - recall--ln = log e
9.) 3 log x + 1/2 log y
try...
11.) 64 - log2 16
Feb 13­2:39 PM
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Logandexponentspart2.notebook
December 16, 2013
C.) Algebra
Express y in terms of x
1.) ln y - ln x = 2 ln 7
2.) ln y = 1 ( ln 4 + ln x)
3
3.) log y + 1 log x = log 3
2
4.) log y = 2 x - 1
Feb 24­9:38 AM
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Logandexponentspart2.notebook
December 16, 2013
Day 5
Graphing Logarithm Functions
1.) Graph y = log3 x
Don't forget topics such as
1.) domain
2.) range
3.) asymptote
4.) inverse
Jan 4­1:58 PM
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2.) Graph y = log5 (x-2)
What if I had to graph ...
y = log5 (x-2) + 4
Notice any changes????
Jan 4­1:58 PM
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December 16, 2013
Day 6
Solve for x:
1.) log 2x + log 5 = log 60
2.) log ( x - 4) + log ( x + 4) = log 9
3.) log ( x + 21) + log x = 2
4.) log 5 ( x + 3) - log 5 x = log 5 4
Feb 24­9:42 AM
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December 16, 2013
day 8
Log Equations using the Power Property
old
3x = 81
new
3x = 80
1.) Solve to the nearest tenth: 4x = 81
2.) 3(7 x )= 201
3.) ex = 90
* can use short cut
Feb 24­9:45 AM
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4.) ( ex ) 2 = 71
5.) 2 2x - 2x - 6 = 0
hint...use a substitution let y = 2x
6.) e 2x - 5 ex + 6 = 0
what substitution can we make?
Feb 24­9:50 AM
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December 16, 2013
Day 9
Word Problems
A(t) = A 0 ( 1 + r) t
where
A0 represents the initial time
t represents time
r represents growth
A(t) represents the amount at time t
Jan 7­1:47 PM
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Examples
1. Suppose the population of a small nation grows at a
rate of 3% per year. In the year 1990, the population
of the nation was 30,000,000.
a. What will the population be now in 2014?
b.) When will the population of the nation be double of 1990?
2.) The value of your $4000 car depreciates at a
rate of 15%. Write an equation that can be used
to determine the value of your can in t years.
Feb 13­12:11 PM
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December 16, 2013
3.) Suppose you invested $1000 in an account such that
A(t) = 1000 2 t/10 where t is years.
a.) How much money did you invest?
b.) How long does it take for your money to triple?
4.) The mass m kg of a radio-active substance at
time t hours is given by
m = 4e–0.2 t.
(a) Write down the initial mass.
(b) The mass is reduced to 1.5 kg. How long
does this take?
Feb 13­12:12 PM
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