1. Logarithmic Functions
OBJECTIVE
Convert between logarithmic and exponential equations.
Solve exponential equations.
Solve problems involving exponential and logarithmic functions.
Differentiate functions involving natural logarithms.
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2. 3.2 Logarithmic Functions
DEFINITION:
A logarithm is defined as follows:
The number is the power y to which we raise a
to get x. The number a is called the logarithmic base.
We read as “the logarithm, base a, of x.”
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3. 3.2 Logarithmic Functions
Example 1: Graph:
First we write the equivalent exponential equation:
Then, we will select some values of y to find the
corresponding values of x. Next, we will plot these
points, keeping in mind that x is still the first
coordinate, and connect the points with a smooth curve.
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4. 3.2 Logarithmic Functions
Example 1 (concluded):
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5. 3.2 Logarithmic Functions
THEOREM 3: Properties of Logarithms
For any positive numbers M, N, and a, and any real
number k:
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6. 3.2 Logarithmic Functions
THEOREM 3 (concluded):
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7. 3.2 Logarithmic Functions
Example 2: Given
find each of the following:
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8. 3.2 Logarithmic Functions
Example 2 (continued): e ) l o g a = 1 2 = 1 2 l o g a = 1 2
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9. 3.2 Logarithmic Functions
Example 2 (concluded):
we cannot find:
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10. 3.2 Logarithmic Functions
Quick Check 1
Given and , find each of the following:
a.) b.) c.)
d.) e.) f.)
a.)
b.)
c.)
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11. 3.2 Logarithmic Functions
Quick Check 1 Concluded
d.)
e.)
f.)
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12. 3.2 Logarithmic Functions
DEFINITION:
For any positive number x,
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13. 3.2 Logarithmic Functions
THEOREM 4: Properties of Natural Logarithms
SAME AS log x
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14. 3.2 Logarithmic Functions
Example 3: Solve for t.
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15. 3.2 Logarithmic Functions
THEOREM 5
ln x exists only for positive numbers x. The domain is
(0, ∞).
ln x < 0 for 0 < x < 1.
ln x = 0 when x = 1.
ln x > 0 for x > 1.
The function given by f (x) = ln x is always increasing.
The range is the entire real line, (–∞, ∞).
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16. 3.2 Logarithmic Functions
THEOREM 6
For any positive number x,
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17. 3.2 Logarithmic Functions
Quick Check 2
Differentiate:
a.) ,
b.) ,
c.) ,
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18. 3.2 Logarithmic Functions
THEOREM 7 or
The derivative of the natural logarithm of a function is
derivative of the function divided by the function.
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19. 3.2 Logarithmic Functions
Example 4: Differentiate
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20. 3.2 Logarithmic Functions
Example 4 (concluded):
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21. 3.2 Logarithmic Functions
Quick Check 3
Differentiate:
a.)
b.)
c.)
d.)
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22. 3.2 Logarithmic Functions
Example 5: In a psychological experiment, students
were shown a set of nonsense syllables, such as POK,
RIZ, DEQ, and so on, and asked to recall them every
second thereafter. The percentage R(x) who retained
the syllables after t seconds was found to be given by
the logarithmic learning model:
a) What percentage of students retained the syllables after 1 sec?
b) Find R(2), and explain what it represents.
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23. 3.2 Logarithmic Functions
Example 5 (concluded): a) b)
The result indicates that 2 seconds after students have
been shown the syllables, the percentage of them who
remember the syllables is shrinking at the rate of
13.5% per second.
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24. 3.2 Logarithmic Functions
Section Summary
A logarithmic function is defined by , for and
The common logarithmic function is defined by , for
The natural logarithm function is defined by , where The derivative of is , for The slope of a tangent line to the graph of at is found by taking the reciprocal of the input .
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