Download Natural Logarithms

Introduction
Logarithms can be used to solve exponential equations
that have a variable as an exponent. In compound
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interest problems that use the formula A = P 1+
,
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logarithms can be used to solve for t, which indicates the
amount of time an account or investment takes to mature.
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4.3.2: Natural Logarithms
Introduction, continued
A financial advisor at a business can use this information
to forecast and plan for the company’s future. In natural
exponential equations (those with the variable e), a
natural logarithm can be used to solve for a variable that
exists as an exponent, since ln x = loge x. So, natural
logarithms can be used to model situations, graphed as
functions, and written as exponents. Additionally, they
can be evaluated by hand using the properties of
logarithms (e.g., ln 1 = 0 because e0 = 1), and with a
calculator when properties of logarithms cannot be
applied. For example, ln 8 ≈ 2.07 because e2.07 ≈ 8.
4.3.2: Natural Logarithms
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Key Concepts
• Recall that natural logarithms consist of a number with
a base of e. As with common logarithms, natural
logarithms have an inverse relationship with
exponential functions. That is, the natural logarithm
function ln b = a can also be rewritten as the
exponential function, ea = b, where e is an irrational
number with an approximate value of 2.71828.
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4.3.2: Natural Logarithms
Key Concepts, continued
• Natural logarithms differ from common logarithms
because of their base. Common logarithms have a
base of 10 whereas natural logarithms have a base of
e. For convenience, natural logarithms can be
converted to base 10 by using the change of base
log x
formula, log x =
.
b
logb
• The same properties that apply to common logarithms
also apply to natural logarithms.
4.3.2: Natural Logarithms
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Key Concepts, continued
• One application for natural logarithms involves
calculating interest on money that is compounded
continuously, or at every instant.
• The continuously compounded interest formula is
A = Pert, where A is the ending amount, P is the
principal or initial amount, e is a constant, r is the
annual interest rate expressed as a decimal, and t is
the time in years. Note this formula’s similarity to the
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compound interest formula, A = P 1+
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4.3.2: Natural Logarithms
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Key Concepts, continued
• The continuously compounded interest formula,
A = Pert, is a natural exponential function because it is
in the form f(x) = ex and involves a base of e.
• Other uses of natural logarithms include situations of
exponential growth or decay, which involve rapid
increase (growth) or decrease (decay).
• One formula for modeling the exponential growth or
decay of a substance or population is P = P0ekt. In this
formula, P represents the amount of a substance or
population after the time, t, has elapsed, P0 is the initial
amount, e is a constant, and k is the rate of growth or
decay.
4.3.2: Natural Logarithms
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Key Concepts, continued
• Notice this formula is similar to the continuously
compounded interest formula, but instead of modeling
an amount of money over time, this formula models the
amount of a substance or population over time. When
k > 0, the formula models growth; when k < 0, the
formula models decay. Since this formula uses the base
e, it represents a natural exponential function.
• A common application of exponential decay is finding
the half-life of a substance. Half-life is the time it takes
for a substance that is decaying exponentially to
decrease to 50% of its original amount.
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4.3.2: Natural Logarithms
Common Errors/Misconceptions
• confusing the formula for continuously compounded
interest with that for compound interest and vice versa
• making substitution errors when converting exponential
functions to logarithmic functions and vice versa
• misinterpreting the parts of an exponential function
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4.3.2: Natural Logarithms
Guided Practice
Example 1
Cheyenne deposited $200 in a bank account earning
continuously compounded interest. After 10 years, she
closed the account and withdrew the entire balance,
which totaled $364.42. What was her annual interest
rate? Rounded to the nearest dollar, how much would
Cheyenne have received if she had left the money in the
account for 15 years? 20 years? Use the continuously
compounded interest formula, A = Pert, where A is the
ending amount, P is the principal or initial amount, e is a
constant, r is the annual interest rate expressed as a
decimal, and t is the time in years.
4.3.2: Natural Logarithms
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Guided Practice: Example 1, continued
1. Determine values for the continuously
compounded interest formula.
The continuously compounded interest formula is
A = Pert.
The ending amount, A, is the account balance when
Cheyenne withdrew the money, $364.42.
The principal, P, is $200.
The time, t, is 10 years.
Therefore, let A = 364.42, P = 200, and t = 10.
4.3.2: Natural Logarithms
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Guided Practice: Example 1, continued
2. Substitute the known values into the
formula and solve for the annual interest
rate, r.
A = Pert
Continuously compounded
interest formula
(364.42) = (200)er (10) Substitute 364.42 for A,
200 for P, and 10 for t.
1.82 = e10r
Divide both sides by 200
and simplify the exponent.
ln 1.82 = ln e10r
Rewrite each side as a
natural logarithm.
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4.3.2: Natural Logarithms
Guided Practice: Example 1, continued
ln 1.82 = 10r ln e
r=
ln 1.82
10 ln e
r ≈ 0.06
Apply the Power Property
for natural logarithms.
Divide to isolate r and then
apply the Symmetric
Property of Equality.
Evaluate each natural
logarithm using a calculator.
The value of r is about 0.06; therefore, the annual
interest rate was 6%.
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4.3.2: Natural Logarithms
Guided Practice: Example 1, continued
3. Determine how much money Cheyenne
would have received if she had left the
money in the account for 15 years.
Use the continuously compounded interest formula,
A = Pert.
The values of P and r remain the same.
The value of t is 15.
Substitute these values into the formula and solve for
the ending amount, A.
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4.3.2: Natural Logarithms
Guided Practice: Example 1, continued
A = Pert
A = (200)e(0.06)(15)
Continuously compounded
interest formula
Substitute 200 for P, 0.06
for r, and 15 for t.
A = 200e0.9
Simplify the exponent.
A ≈ 491.92
Evaluate using a calculator.
If Cheyenne had left her money in the account for 15
years, she would have received about $491.92.
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4.3.2: Natural Logarithms
Guided Practice: Example 1, continued
4. Determine how much money Cheyenne
would have received if she had left the
money in the account for 20 years.
Once again, use the continuously compounded
interest formula, A = Pert.
The values of P and r remain the same.
The value of t is 20.
Substitute these values into the formula and solve for
the ending amount, A.
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4.3.2: Natural Logarithms
Guided Practice: Example 1, continued
A = Pert
A = (200)e(0.06)(20)
Continuously compounded
interest formula
Substitute 200 for P, 0.06
for r, and 20 for t.
A = 200e1.2
Simplify the exponent.
A ≈ 664.02
Evaluate using a calculator.
If Cheyenne had left her money in the
account for 20 years, she would have
received about $664.02.
4.3.2: Natural Logarithms
✔
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Guided Practice: Example 1, continued
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4.3.2: Natural Logarithms
Guided Practice
Example 3
Studies have shown that the number of bacteria on a
public restroom sink can grow exponentially from 5,000
bacteria to 12,000 bacteria in 10 hours. Write the natural
logarithmic equation that would represent how the number
of bacteria would grow over the given time period. Use the
exponential growth/decay formula, P = P0ekt, where P
represents the number of bacteria after t hours, P0 is the
initial number of bacteria, and k is the rate of growth or
decay. Given the rate of bacterial growth, how many
bacteria would there be after 24 hours? How many bacteria
would you expect to be present after 48 hours? Verify your
answers algebraically.
4.3.2: Natural Logarithms
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Guided Practice: Example 3, continued
1. Determine the growth rate, k, using
properties of logarithms.
Begin by identifying the known values.
Let P, the final population of the bacteria, be 12,000.
Let P0, the initial population of the bacteria, be 5,000.
Let t, the time in hours, be 10.
Substitute these values into the given formula and
solve for k.
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
P = P0ekt
Exponential growth/decay
formula
(12,000) = (5000)ek (10) Substitute 12,000 for P,
5,000 for P0, and 10 for t.
2.4 = e10k
Divide both sides by
5,000 and simplify the
exponent.
ln 2.4 = ln e10k
ln 2.4 = 10k ln e
4.3.2: Natural Logarithms
Rewrite each side as a
natural logarithm.
Apply the Power Property
of natural logarithms.
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Guided Practice: Example 3, continued
k=
ln 2.4
10 ln e
k ≈ 0.088
calculator.
Divide to isolate k and
then apply the Symmetric
Property of Equality.
Evaluate using a
The growth rate, k, is approximately 0.088.
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
2. Determine the number of bacteria after
24 hours.
Use the exponential growth/decay formula, P = P0ekt.
The value for P0 remains the same (5,000).
Let the growth rate, k, be 0.088.
Let the time, t, be 24.
Substitute these values into the given formula and
solve for P, the final population.
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
P = P0ekt
Exponential growth/decay
formula
P = (5000)e(0.088)(24)
Substitute 5,000 for P0,
0.088 for k, and 24 for t.
P = 5000e2.112
Simplify the exponent.
P ≈ 41,324
Evaluate using a
calculator.
After 24 hours, there will be approximately 41,324
bacteria on the sink.
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
3. Determine the number of bacteria after
48 hours.
Use the exponential growth/decay formula, P = P0ekt.
The values for P0 and k remain the same
(5,000 and 0.088, respectively).
Let the time, t, be 48.
Substitute these values into the given formula and
solve for P, the final population.
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
P = P0ekt
Exponential growth/decay
formula
P = (5000)e(0.088)(48)
Substitute 5,000 for P0,
0.088 for k, and 48 for t.
P = 5000e4.224
Simplify the exponent.
P ≈ 341,531
Evaluate using a
calculator.
After 48 hours, there will be approximately
341,531 bacteria on the sink.
✔
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4.3.2: Natural Logarithms
Guided Practice: Example 3, continued
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4.3.2: Natural Logarithms