EXAMPLE 7 Solve e 2x – 3e x + 2 = 0

WORKSHEET KEY
2
1/11/2017 11:53 PM
c
5.5 - Properties of Logarithms
1
NATURAL
LOGARITHMS
Section 5.5A,
Revised ©2012,
[email protected]
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
2
CONVERSIONS
A. The base of Natural Log (ln) is e.
B. Convert: lne x = y to ey = x
1/11/2017 11:53 PM
5.5 – Properties of Logarithms
3
EXAMPLE 1
Convert ln 2/5 = –0.916 to exponential form
ln e x  y
y
e =x
e
1/11/2017 11:53 PM
0.916
2

5
5.5 – Properties of Logarithms
4
YOUR TURN
Convert ln 679 = 6.520 to exponential form
e
1/11/2017 11:53 PM
6.520
 679
5.5 – Properties of Logarithms
5
EXAMPLE 2
Convert e2 = 7.389 to logarithmic form
e y
x
ln y  x
ln7.3890  2
1/11/2017 11:53 PM
5.5 – Properties of Logarithms
6
YOUR TURN
Convert e2x = 3 to logarithmic form
ln3  2x
1/11/2017 11:53 PM
5.5 – Properties of Logarithms
7
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
1/11/2017 11:53 PM
5.5 – Properties of Logarithms
8
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
1/11/2017 11:53 PM
e
5.5 - Properties of Logarithms
ln x
9
EXAMPLE 3
Solve ln x = 4
ln x  4
ln x 
e
4
x  54.598
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
10
EXAMPLE 4
Solve 5 + 2 ln x = 7
x  2.718
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
11
YOUR TURN
Solve 3 ln x – 6 = 9
x  148.413
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
12
EXAMPLE 5
Solve ex + 5 = 60
e  5  60
x
e  55
x
ln e  ln 55
x
x  4.007
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
13
EXAMPLE 6
Solve –14 + 3ex = 11
x  2.120
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
14
YOUR TURN
Solve 7 – 2ex = 5
x0
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
15
REVIEW
Solve x2 – 3x + 2 = 0
x  3x  2  0
2
 x  2 x  1  0
1, 2
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
16
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
1/11/2017 11:53 PM
x
5.5 - Properties of Logarithms
17
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
1/11/2017 11:53 PM
x
 0.693,0
5.5 - Properties of Logarithms
18
EXAMPLE 8
Solve e2x – 7ex + 12 = 0
 1.099,  1.386
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
19
EXAMPLE 9
Solve 2e2x + 7ex – 4 = 0
 0.693ext : ln  4
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
20
YOUR TURN
Solve 6e2x + 11ex – 2 = 0
 1.792ext : ln  2
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
21
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
1/11/2017 11:53 PM
1000

500
5.5 - Properties of Logarithms
22
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
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
23
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
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
24
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
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
25
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
1/11/2017 11:53 PM
P0== Initial
5,000 Amount
e = The Natural Base
kK = ??
Growth or Decay
Rate
t = 10
T = Time
5.5 - Properties of Logarithms
26
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
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
27
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?
1/11/2017 11:53 PM
k  0.0926
5.5 - Properties of Logarithms
28
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.
1/11/2017 11:53 PM
P0  65,050
5.5 - Properties of Logarithms
29
ASSIGNMENT
Worksheet
1/11/2017 11:53 PM
5.5 - Properties of Logarithms
30