Checklist and Assignment
Notes
Checklist:
Logarithms
Exact Doubling Time and Half-Life
Assignment:
1. p 487 - 488 Quick Quiz # 8 - 10
2. p 488 - 490 Exercises 13 - 24, 53, 55, , 63, 64
An investment growing exponentially will double in about
15 years. When will the investment triple?
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Key Words
Notes
Logarithms or logs, are exponents. We focus on
”base 10” logs.
log10 x means, ”10 to which power equals x?”
For example, log10 1000 = 3 because 103 = 1000.
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Logarithms - Meaning
Notes
log10 x is the power to which 10 must be raised to obtain
x.
More examples:
log10 10, 000, 000 = 7 means 107 = 10, 000, 000
log10 1 = 0
means 100 = 1
log10 0.1 = −1
means 10−1 = 0.1
Basically any real number can be an exponent of ten.
log10 25 ≈ 1.3979 because 101.3979 ≈ 25
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Logarithms - Rules
Notes
Logarithm rules follow from the properties of exponents.
1. Taking the logarithm of a number that is a power of 10
gives the power.
log10 10x = x
2. Raising 10 to a number that is a logarithm of x gives
back x. x > 0.
10log10 x = x
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Logarithms - Rules
Notes
3. Because powers of 10 are multiplied adding the powers,
the addition rule of logarithm is as follows:
log10 x · y = log10 x + log10 y .
4. We can ”bring down” a power within a logarithm using
the power rule for logarithms:
log10 ax = x · log10 a
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Use of Logarithm Rules
Notes
The use of the four rules can help with specific problems.
We are given log10 2 ≈ 0.301030. Check this on your
calculator.
Find log10 4.
Find log10 8.
Find log10 200.
Find 10log10 2 .
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Exact Doubling Time and Half-Life
Notes
Now with logarithms, we can get exact formulas for the
doubling time and half-life
Given a percentage growth rate P%, the fractional growth
rate r = P/100. When P% is a decay rate, r = −P/100.
Tdouble =
Thalf =
log10 2
log10 (1 + r )
log10 2
log10 (1 + r )
When P is growth, r > 0. When P is decay, r < 0.
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Exercises
Notes
Given log10 5 ≈ 0.699 find each without a calculator.
1. log10 50
2. log10 0.05
3. log10 25
4. log10 0.20
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Exercises
Notes
Inflation is causing prices to rise at a rate of 12% per
year. For an item that costs $500 today, what will be the
price in 4 years.
Compare the doubling times using the rule of 70
(approximate) and logarithms (exact). Then use the exact
doubling time to answer the question.
When will the price of the item be worth $1500? (When
will it triple?)
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