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WORKSHEET KEY
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5.5 - Properties of Logarithms
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NATURAL
LOGARITHMS
Section 5.5A,
Revised ©2012,
viet.dang@humble.k12.tx.us
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5.5 - Properties of Logarithms
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CONVERSIONS
A. The base of Natural Log (ln) is e.
B. Convert: lne x = y to ey = x
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EXAMPLE 1
Convert ln 2/5 = –0.916 to exponential form
ln e x  y
y
e =x
e
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0.916
2

5
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YOUR TURN
Convert ln 679 = 6.520 to exponential form
e
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6.520
 679
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EXAMPLE 2
Convert e2 = 7.389 to logarithmic form
e y
x
ln y  x
ln7.3890  2
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YOUR TURN
Convert e2x = 3 to logarithmic form
ln3  2x
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SIMPLIFYING NATURAL LOGARITHMS
A. The inverse of a natural base (e) is the
natural log (ln)
B. If there is ln and e, they cancel each other
out
C. Natural logarithms have the same properties
as log base 10 and logarithms with other
bases
D. The base of a natural log is e but it will never
be written as the base.
E. ln + e = FELONY
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NATURAL LOGARITHM STEPS
A. RAISE IT UP by incorporating e as the base to
both sides
B. Cancel any ln e ’s
C. Simplify using Natural Logarithm rules
D. Check
ln e x
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e
5.5 - Properties of Logarithms
ln x
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EXAMPLE 3
Solve ln x = 4
ln x  4
ln x 
e
4
x  54.598
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EXAMPLE 4
Solve 5 + 2 ln x = 7
x  2.718
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YOUR TURN
Solve 3 ln x – 6 = 9
x  148.413
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EXAMPLE 5
Solve ex + 5 = 60
e  5  60
x
e  55
x
ln e  ln 55
x
x  4.007
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EXAMPLE 6
Solve –14 + 3ex = 11
x  2.120
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YOUR TURN
Solve 7 – 2ex = 5
x0
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REVIEW
Solve x2 – 3x + 2 = 0
x  3x  2  0
2
 x  2 x  1  0
1, 2
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EXAMPLE 7
Solve e2x – 3ex + 2 = 0
e  3e  2  0
2x
x
 e   3e
 e  2  e
x 2
x
20
x
x
 1  0
e 20
e 1  0
x
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x
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EXAMPLE 7
Solve e2x – 3ex + 2 = 0
e 20
x
e 2
x
ln e  ln 2
x ln e  ln 2
x  ln 2
x  0.693
e 1  0
x
e 1
x
ln e  ln1
x ln e  ln1
x  ln1
x0
x
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x
 0.693,0
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EXAMPLE 8
Solve e2x – 7ex + 12 = 0
 1.099,  1.386
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EXAMPLE 9
Solve 2e2x + 7ex – 4 = 0
 0.693ext : ln  4
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YOUR TURN
Solve 6e2x + 11ex – 2 = 0
 1.792ext : ln  2
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EXAMPLE 10
You have deposited $500 in an account that pays 6.75%
interest, compounded continuously. How long will it take
your money to double?
rt
Doubled
Amount
A  Pe
0.0675t
A  500e
0.0675t
1000  500e
0.0675t
500e
 1000
0.0675t
500e
500
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1000

500
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EXAMPLE 10
You have deposited $500 in an account that pays 6.75%
interest, compounded continuously. How long will it take
your money to double?
rt
A  Pe
0.0675 t
e
2
0.0675t
ln e
 ln 2
0.0675t  ln 2
ln 2
t
0.0675
 10.269 years
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EXAMPLE 11
You have deposited $2,500 in an account that pays 8.5%
interest, compounded continuously. How long will it take
your money to triple?
 12.925years
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YOUR TURN
How long will it take $30,000 to accumulate to $110,000 in a
trust that earns a 10% annual return compounded
continuously?
 12.993years
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EXAMPLE 12
During its exponential growth phase, a certain bacterium can
grow from 5,000 cells to 12,000 cells in 10 hours. What is the
growth rate?
kt
P = 12,000
Ending Amount
0
P  Pe
 k 10
12,000  5,000e
12,000  k 10
e
5,000
12
 k 10 
ln  ln e
5
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P0== Initial
5,000 Amount
e = The Natural Base
kK = ??
Growth or Decay
Rate
t = 10
T = Time
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EXAMPLE 12
During its exponential growth phase, a certain bacterium can
grow from 5,000 cells to 12,000 cells in 10 hours. What is the
growth rate?
12
 k 10 
ln  ln e
5
12
ln  10k ln e
5
 12 
ln  
 5  k
10
k  0.0875
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EXAMPLE 13
During its exponential growth phase, a certain bacterium can
grow from 5,000 cells to 15,000 cells in 12 hours. What is the
growth rate?
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k  0.0926
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YOUR TURN
The population of a certain city in 2000 was 99,500. What is its
initial population in 1975 when its growth rate is at 0.170.
Round to the nearest whole number.
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P0  65,050
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ASSIGNMENT
Worksheet
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