6.2 Logarithmic Functions
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Transcript of 6.2 Logarithmic Functions
8/4/2019 6.2 Logarithmic Functions
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Transcendental
Functions6.2 Logarithmic Functions
RA Idoy
MATH17
8/4/2019 6.2 Logarithmic Functions
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• Logarithmic Function with base
b
• Graph of Logarithmic Function
• Natural Logarithmic Function
• Properties of Logarithmic
Functions
• Exponential and Logarithmic
Equations
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Logarithmic Function with
base b
Definition:
The logarithmic function with
base b is the inverse of the
exponential function with base
b.
Note:
log if and only if
y
b y x x b
Dom f Rng f
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Logarithmic Function with
base b
Examples:
2
33 9 log 9 2
1/2
1/16
1 1 1 1log
16 4 4 2
32 8
2 15
25
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Logarithmic Function with
base b
More examples:
Find the values of the ff:
Solution:
Therefore,
7log 497
2
log 49
7 497 7
y
y
y
7log 49 2
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Logarithmic Function with
base b
More examples:
Find the values of the ff:
Solution:
Therefore,
5log 55
1/2
log 5
5 5
5 5
y
y
y
5log 5 1/ 2
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Logarithmic Function with
base b
More examples:
Find the values of the ff:
Solution:
Therefore,
6
1
log 66
1
log 1/ 6
6 1/ 6
6 6
y
y
y
6log 1/ 6 1
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Logarithmic Function with
base b
More examples:
Find the values of the ff:
Solution:
Therefore,
3log 813
4
log 81
3 81
3 3
y
y
y
3log 81 4
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Logarithmic Function with
base b
More examples:
Find the values of the ff:
Solution:
Therefore,
10log 0.001
10
3
log 0.001
10 0.00110 10
y
y
y
10log 0.001 3
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Logarithmic Function with
base b
Other examples:
Solve the given equation for either
x or b.
Solution:
6log 2 x
6
2
log 2
6
36
x
x
x
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Logarithmic Function with
base b
Other examples:
Solve the given equation for either
x or b.
Solution:
27log 2 / 3 x
27
2/3
23
log 2 / 3
27
27
9
x
x
x
x
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Logarithmic Function with
base b
Other examples:
Solve the given equation for either
x or b.
Solution:
log 4 1/ 3b
1/3
31/3 3
log 4 1 / 3
4
4
64
b
b
b
b
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Logarithmic Function with
base b
Other examples:
Solve the given equation for either
x or b.
Solution:
log 81 2b
2
1/22 1/2
log 81 2
81
81
1/ 9
b
b
b
b
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Logarithmic Function with
base b
Recall:
Since the logarithmic function with
base b is the inverse of exponential
function with base b, then
1
1
f f x x
f f x x
log
log
b
xb
x
b x
b x
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Logarithmic Function with
base b
Example:
5
2log 2 5
3
10log 10 3
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Graph of Logarithmic
function of base b
Since logarithmic function is
the inverse of exponential
function, then its graph is the
reflection of the latter
function at y=x.
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x y b
0b
logb
y x
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x y b
0 1b
logb
y x
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Natural Logarithmic
Function
Definition:
The natural logarithmic
function is the inverse of the
natural exponential function.
ln if and only if y y x x e
Dom f Rng f
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Natural Logarithmic
Function
Note:
ln 1e ln x
e x
lnx
e x
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Natural Logarithmic
Function
Example 1: Recall the function
obtained from exponential
growth of bacteria culture.
Determine how many minutes
elapse until there are 30,000bacteria present in the
culture.
0.041500
t f t e
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Natural Logarithmic
Function
Solution:
Let T be the
time elapse
until there are30,000 bacteria
present
Given:
0.041500 T f T e
30,000 f T 0.04
0.04
0.04
30000 150020
ln 20 ln
ln 20 0.04
74.9
T
T
T
ee
e
T
T
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Natural Logarithmic
Function
Conclusion:
Therefore 74.9 minutes or 1
hr, 14 min and 54 sec elapse
until there are 30,000 bacteria
present.
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Natural Logarithmic
Function
Example 2:
If $1000 is deposited into a
savings account that pays an
annual interest rate of 6percent compounded
continuously, and no
withdrawals or additional
deposits are made, how long
will it take until there is
$1500 on deposit?
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Natural Logarithmic
Function
Solution:
Let T years
be the time to
take toaccumulate $1500
Recall: it A Pe
0.061500 1000
T e
0.06
0.06
0.06
1500 1000
1.5ln1.5 ln
ln1.5 0.066.758
T
T
T
e
e
e
T T
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Natural Logarithmic
Function
Conclusion:
Therefore it takes 6.758
years or 6 years, 9 months and
3 days for the initial deposit
to accumulate to $1500.
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Properties of Logarithmic
Functions
Theorem:
If b>0, b≠1, and u and v are
positive numbers, then
log log logb b b
uv u v
log log logb b b
u u vv
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Properties of Logarithmic
Functions
Theorem:
If b>0, b≠1, and u and v are
positive numbers, then
log logn
b bu n u
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Properties of Logarithmic
Functions
Examples: Express each of the
following in terms of log of x, y
and z, each of which represents a
positive number.2 3 4
logb
x y z
2log
b
x
yz
2
5
3log
b
xy
z
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Properties of Logarithmic
Functions
Examples: Write each of the
following expression as a single
logarithm with a coefficient of 1.
log 2log 3logb b b
x y z
1 log 4 log 3 log log3
b b b b x y
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Exponential and
Logarithmic Equations
Examples: Find the solution set.
1.
2.
3.
4.5.
3 16
x 3 1
5 0.08 x
17 3
x x
4log 19 10
log 3 2 x
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Exponential and
Logarithmic Equations
Examples: Find the solution set.
6.
7.
8.
2 2
log 4 log 3 3 x x
3 3log log 2 3 3 x x
3 3 4 x x
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Exponential and
Logarithmic Equations
Suppose on January 1, 1988, the
population of a certain city was
800,000. From then until the year
2000 the population is expected
to increase at the rate of 3.5
percent per year. Therefore t
years after January 1, 1988, the
population is expected to be800,000(1.035)t, where 0≤t≤12.When would you predict the
population will be one million?
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Exponential and
Logarithmic Equations
• We wish to
determine the
value of t
800000 1.035 1000000
1.035 1.25
t
t
log 1.035 log1.25
log 1.035 log1.25log1.25
log1.035
6.5
t
t
t
t
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Exponential and
Logarithmic Equations
Conclusion:
Therefore, after six and one-
half year, the population will
be one million. This is on July
1, 1994.
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Exponential and
Logarithmic Equations
If $1000 is deposited in a
savings account that pays an
annual interest rate of 6
percent compounded quarterlyand no withdrawals or
additional deposits are made,
how long will it take untilthere is $1500 on deposit?
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Exponential and
Logarithmic Equations
0.061500 1000 1
4
n
1.5 1.015
log1.5 log 1.015
log1.5 log 1.015
log1.5
log1.01527.23
n
n
n
n
n
1
n
n
i A P
m
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Exponential and
Logarithmic Equations
Conclusion:
Therefore, it will take 28
quarters until there is $1500
on deposit.