Logarithms with Timon and Pumba

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Logarithms with Timon and Pumba By Anna
Simanovskaia
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The Definition
Timon, What is a logarithm?!?!?!
The log of a number a to the base b is the
exponent c to which the base b must be raised to
get number a.
What in the world?
For all b0, b?1, and a0 we can say Logbac Is
an alternative way of writing abc
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Definition 1
What is it? I mean, its not a fallen tree with
bugs, is it?
Pumba, there are two definitions. 1) b(logba)
a- means the exponent to which b must be raised
to yield a. Therefore, if b is raised to this
power the answer must be a.
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Definition 2
Oh, I think I am getting this. What is the next
definition?
Finally, Okay. 2) Logbbaa means that the
exponent to which b is raised, when the base is
also b, is equal to the log function.
By the way Pumba, the two definitions that I just
told you are called fundamental logarithmic
identities.
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Laws of logarithms
Mhm, wait, now are there laws?
Yup. They help you solve the log problems. So law
1 Logb10 Exponent form is b01
Ex. Logb1loga1/logab0/logab0
Log101log21/log2100/log2100
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Laws of logarithms
This is kinda cool. What is the next law Timon,
huh, huh, huh?
Thank you Pumba. Law2 Logbb1 Exponent- b1 b
Ex. Logbblogab/logab1 Log1010 loga10/loga101
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Law of logarithms
Hey Timon do you think that you can tell me
another?
Yeah sure Pumba. Log Law 3 is
logb(nm)logbnlogbm the log of a product is
equal to the sum of the logs of the factor bnbm
bnm
Ex. Log2(23)log22log23
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Law of logarithms
Pumba is so interested in logs. How should I say
the next one? How about the next law
is Logb(n/m) logbn-logbm.
Wow I think I could get some food
Ex. Log4(6/3)log46-log43
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Law of logarithms
This is a great place, do you have anything to
say Timon.
Yes I do Pumba. Logb(nm)mlogbn the log of a
number to a power is the exponent times the log.
(bn)mbnm
Ex. Log2(34)4log23
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Law of logarithms
Timon I can not take it anymore, how many more
laws are there?
There is only one more.
Logbkm 1/k logbm the power of the log is the
product of the reciprocal of the power and the
log
Ex. Logbkmlogm/logbklogm/klogb(1/k)logbm Log23
41/3log24
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Applications
The universal model f(x) Ce (-kx)

• Radioactive decay
• The amount of a radioactive substance in an
  object decreases exponentially with respect to
  time
• You can use this relationship to date a fossil
  (if you know the initial amount of a radioactive
  substance as well as the current amount) or to
  find the half-life (usually known)

f(x) amount of substance now C initial amount
of substance k ln(2) / half life x time passed
Example The half-life of C-14 is 5730 years. If
the amount of C-14 left in the human is 25 of
the initial amount, how much time has passed
since he died?
Answer f(x)/Ce(-ln(2)x/half-life) ¼e(ln(2(-
x/half-life))) ¼2(-x/5730) ln(1/4)(-x/5730)ln(2
) x -5730ln(¼)/ln(2) 11460 years
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The universal model f(x) Ce (-kx)
Law of cooling (Newton) The temperature of an
object decreases exponentially over time.
For cooling, the model varies a little T(t) S
(R S)e(-kt) where T(t) current
temperature S surrounding temperature R
initial temp. of object t time passed k a
constant
Example If room temperature is 20 degrees, the
temperature of a soup is 100 degrees at t0, and
it cools down to 60 degrees in 10 minutes, how
long will it take to cool down to 40 degrees?
Answer T(t) S (R S)e(-kt) 6020 (100
20)e(-k10) k -ln(½)/10 .0693 4020 (100
20)e(-.0693t) ln(¼) ln(e(-.0693t)) t
-ln(¼)/.0693 20 minutes
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The universal model f(x) Ce (-kx)

• Continuously compounded interest
• When you put money in the bank, you get money
  either every year (annual interest), month
  (monthly), 3 months (quarterly), week (weekly),
  day (daily), or continuously (all the time).
• The amount of money in the bank increases
  exponentially.

The model for continuously compounded interest
also differs from the universal model a bit
(since it is increasing) f(x) Ce (kx) f(x)
current balance C initial amount of money x
time in bank (in years) K interest rate
Example If Kristina puts 10000 in a bank for
her college education savings, how much will she
have in 5 years if the bank gives her 7 interest
compounded continuously?
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The Mathematical Constant e
Hey Timon, you keep talking about e. Does it have
something to do with eating?
Ummnot reallyIts kind of complicated. However,
I can tell you that its an irrational number
(like p) equal to 2.7182818
Euler defined the number as lim(x?8) (1
1/x)x (Limit as x goes to infinity of (11/x) to
the x power)
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Property of logarithms
I thought that you said there were no more laws?!
There are no more laws, but there are two
properties.
We have properties, which allow us to change the
base of logarithms.
logbmlogam/logab the changing of the base
property will help you a lot when you need to use
a calculator. If you look at it youll notice
that on a calculator you will have ln key and
log key.
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Property of logarithms
Definition of common logarithms Log alog10a and
is known as a common logarithm (logarithm to the
base of 10) Definition of Natural Logarithms In
alogea and is known as natural logarithms
(logarithms to the base of e)
I can not take it Timon my brain hurts. There is
a lot of stuff to remember.
Its the last one Pumba. Logba1/logab
This is a special case of changing the base
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How to solve a logarithmic equation
The equation Log4x3log2x7
log(x)/log(4)3log(x)/log(2)7
log(x)/log(22)3log(x)/log(2)7
log(x)/(2log(2))3log(x)/log(2)7 (log(x)
6log(x))/2log(2)7 log(x)6log(x)72log(2) 7log(x
)14log(2) log(x)2log(2) log(x)log(22) x4
The equation Log2(x1)-log2(x-1)1
or
logxlogxlog10000 logxlogxlog104 log(x)log(x)4l
og10 log(x)24 log(x)2 102 x 100x
Log2(x1/x-1)1 log(x1/x-1)/Log21 Log(x1/x-1)
log2 x1/x-12 x12x-2 -x-3 X3
the equation Xlogx10000
or
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Different strategies of solvingexponential
equations and inequalities
The inequality equation Log0.5(3x-2)-log2(3x-2)1 log2(3x-2)log22
(3x-2)2 3x4 x4/3 3x-20 3x2 x2/3
The exponential equation 5x100
log5xlog100log5xlog102 Log5x2log10 Log5x2
xlog52 X2/log5
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THE END