Transcript LOGARITHMS Mrs. Aldous, Mr. Beetz & Mr. Thauvette IB DP SL Mathematics.

LOGARITHMS
Mrs. Aldous, Mr. Beetz & Mr. Thauvette
IB DP SL Mathematics
You should be able to…
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Use logarithms to solve exponential equation with
different bases
Apply the laws of logarithms to simplify expressions
Find a logarithm to any base by using the change
of base formula
Solve equations involving logarithms
Investigate
1.
a) The number of bacteria in a culture starts at 1, and
doubles every hour. Let y represent the number of
bacteria, and x represent the time, in hours. Make a
table of values relating x and y.
b) What exponential function relates y to x?
2.
To determine when there will be 15 bacteria in the
culture, what equation would have to be solved?
Investigate continued…
3.
a) Graph y = 2x and its inverse on the same set of axes.
Recall that, to find the inverse function, interchange the xand y-values.
b) Describe the graphical relationship between a function
and its inverse, and their relationship to the line y = x.
c) Explain how the graph of the inverse can be used to
approximate the solution to the equation determined in
step 2.
Investigate continued…
4.
Find an approximate solution for each equation by
carefully graphing an appropriate function and its
inverse.
a) 3x = 12 b) 6x = 17 c) –2x = –9
5.
Make a general statement explaining how to solve
exponential equations graphically.
How can you solve for an unknown
exponent?
Suppose you invest $100 in an account that pays 5%
interest, compounded annually. The amount A, in
dollars, in the account after any given time, t, in years,
is given by A = 100(1.05)t .
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Predict how long it will take, to the nearest year, for
the amount in this account to double in value. Give
reason for your estimate.
A = 100(1.05)
t
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Design a method that will allow you to find an
accurate answer to your estimation.
Carry out your method. How long will it take for the
investment to double in value?
Compare this result with your prediction. How close
was your predication?
A = 100(1.05)
t
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Suppose the initial amount invested is $250. How
does this affect your answer from the last step?
Explain, using mathematical reasoning.
Reflect – How can you express the original equation
A = 100(1.05)t in logarithmic form?
What is the power law for logarithms?
Evaluate each logarithm. Organize your results in a
table.
(i) log 2 (ii) log 4 (iii) log 8 (iv) log 16 (v) log 32
Look for a pattern in your results. How are these vales
related to log 2? Make a prediction for
(i) log 64 (ii) log 1024
Verify your predictions
Write a rule for the general result of log 2n
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Do you think the general result will work for other
powers?
Repeat the analysis for powers of 3:
(i) log 3 (ii) log 9 (iii) log 27
Write a rule for evaluating log 3n.
Verify your rule using a few cases.
Reflect
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Write a rule for evaluating
log b for any base b > 0.
n
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Test your rule using several different cases.
Task – Not Fatal
Viral infections, while often quite severe and requiring
admission to the hospital, are rarely fatal. Most
people recover and go home.
In this task, you will simulate recovery from a viral
infection for 100 people who contract the infection.
Materials needed: http://www.random.org/coins/.
Heads represents a person who has recovered, and
tails represents a person who is still ill.
Task – Not Fatal
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http://www.random.org/coins/
Toss 100 coins.
Count all the coins that turn up heads. These are the
people who have recovered after one week in the
hospital, and get to go home.
Record in a table of values the number of coins
remaining (people still ill) versus the number of
tosses.
Repeat the process with the remaining coins until
you have no coins left.
Task – Not Fatal
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Use technology to construct a scatterplot of the data,
with the number of tosses (days) as the independent
variable and the number of people still ill as the
dependent variable.
Determine an equation for the curve of best fit for your
data. Explain how you determined the best model.
Predict how ling it would take for 1600 people to get
well and go home.
Justify your prediction algebraically.
Why does a logarithmic model work well for this
situation?
For what other situations might this model be
appropriate? Justify your answer.
Question
Given that log5x = y, express each of the following in
terms of y.
(a) log5x2
Question continued…
Given that log5x = y, express each of the following in
terms of y.
1ö
æ
(b) log 5 ç ÷
è xø
Question continued…
Given that log5x = y, express each of the following in
terms of y.
(c) log25x
Practice
Solution
Practice
Solution
Practice
Solution
Practice
Solution
You should know…
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How to think of logarithms as exponents—that is,
logax means “the exponent to which a must be
raised to give x”. For example, log39 means “the
exponent to which 3 must be raised to give 9”,
which is 2. Therefore, log39 = 2.
The laws of logarithms:
log a M + log a N = log a MN
M
log a M - log a N = log a
N
log a M n = n log a M
You should know…
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If it is not possible to make the bases the same, you
can use the properties of logarithms to solve an
exponential equation. For example:
2 =3
x log 2 = log 3
log 3
x=
log 2
x » 1.58
x
Be prepared…
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One of the keys to success in solving logarithmic
equations is your ability to move easily between the
logarithmic form and the exponential form.
Remember that if a = logbx, then x = bn.
The change of base formula allows you to express
any logarithm in terms of another base. This is useful
when graphing logarithmic functions or evaluating
logarithms that do not have base 10 or base e.