Using logarithms
to solve for t
in exponential growth
problems
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Logarithms
• Use exponents as an alternative way to
represent numbers
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Logarithms
• Use exponents as an alternative way to
represent numbers
• Tell the power to which the base
number (usually 10) must be raised to
obtain a value X
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Logarithms
• Alternative way of representing numbers
using exponents
• Tell the power to which the base number
(usually 10) must be raised to obtain a
value X
• log10X signifies the power to which 10
must be raised to equal X
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Logarithms
y
log10X = Y is identical to 10 = X
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An example of logarithms
The log of 100 in base 10
log10100 = 2
because
2
10 =100
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Another example of logarithms
The log of 1000 in base 10
[Log101000] = 3
because
3
10 =100
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A third example
The log of 30 in base 10 [Log1030] = 1.477
because
1.477
10
=30
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Calculate logs in Excel using Log
functions
=Log(number)
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In Excel, compute
1. The log of your age
2. The log of your weight
3. The log of your number of credits [If you
are a first-term freshman, use the
number of credits you will have after this
quarter]
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Key rule for using logarithms:
x
Log a = x * log a
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An example of the use of logs
A Petri dish contains 100 bacteria cells. The
number of cells increases 5% every minute.
How many minutes will it take for there to be
3000 bacteria in the dish?
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An example of the use of logs
Using the simulation method, we can find
that there will be 2898 cells in minute 69 and
3043 in minute 70 but it will not tell us
exactly when there were 3000 cells.
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The example continued
Begin with the formula for exponential
growth:
t
3000 = 100 * 1.05
That is, at 5% growth per minute,
100 cells will grow into 3000
in t minutes.
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The example continued
Step 2: divide both sides by 100,
which gives us:
t
30 = 1.05
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The example continued
Step 3: take the log of each side:
t
Log(30) = log(1.05 )
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The example continued
t
Step 4: apply the rule [log a = t*log a]
log 30 = t* log 1.05
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The example continued
Step 5, divide both sides by log 1.05:
log 30/log 1.05 = x
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The example completed
Step 6, use Excel to compute the value of x.
=log(30)/log(1.05)
69.71 = x
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