3.3 PROPERTIES OF LOGARITHMS - Academics Portal …academics.utep.edu/Portals/1788/CALCULUS...
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3.3 PROPERTIES OF LOGARITHMS
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• Use the change-of-base formula to rewrite and
evaluate logarithmic expressions.
• Use properties of logarithms to evaluate or
rewrite logarithmic expressions.
• Use properties of logarithms to expand or
condense logarithmic expressions.
• Use logarithmic functions to model and solve
real-life problems.
What You Should Learn
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Change of Base
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Change of Base
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Example 1 – Changing Bases Using Common Logarithms
a.
b.
Use a calculator.
Simplify.
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Properties of Logarithms
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Properties of Logarithms
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Example 3 – Using Properties of Logarithms
Write each logarithm in terms of ln 2 and ln 3.
a. ln 6 b. ln
Solution:
a. ln 6 = ln (2 3)
= ln 2 + ln 3
b. ln = ln 2 – ln 27
= ln 2 – ln 33
= ln 2 – 3 ln 3
Rewrite 6 as 2 3.
Product Property
Power Property
Rewrite 27 as 33.
Quotient Property
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Example
Find the exact value of each expression without using a
calculator.
a. log _5 53
= log5 513 =
1
3log5 5 =
1
3
b. ln 𝑒6 − ln 𝑒2
= 6 ln 𝑒 − 2 ln 𝑒 = 6 − 2 = 4
𝑜𝑟 = ln𝑒6
𝑒2= ln 𝑒6−2 = ln 𝑒4 = 4 ln 𝑒 = 4
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Rewriting Logarithmic Expressions
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Example 5 – Expanding Logarithmic Expressions
Expand each logarithmic expression.
a. log4 5x3y
b.
Solution:
a. log4 5x3y = log4 5 + log4 x3 + log4 y
= log4 5 + 3 log4 x + log4 y
Product Property
Power Property
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Example 5 – Solution
b.
Rewrite using rational
exponent.
Quotient Property
Power Property
cont’d
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Example
Condense each logarithmic expression.
a.1
2log 𝑥 + 3 log(𝑥 + 1)
= log 𝑥12 + log 𝑥 + 1 3
= log 𝑥12 𝑥 + 1 3
= log 𝑥 𝑥 + 1 3
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Continue…
b. 2 ln(𝑥 + 2) − ln 𝑥
= ln 𝑥 + 2 2 − ln 𝑥
= ln𝑥 + 2 2
𝑥
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Continue…
c. 1
3[log2x + log2 x + 1 ]
=1
3log2 𝑥(𝑥 + 1)
= log2 𝑥 𝑥 + 113
= log2 𝑥(𝑥 + 1)3
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Application
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Application
If the points are graphed and fall on a line, then you can
determine that the x- and y-values are related by the
equation
ln y = m ln x
where m is the slope of the line.
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Example 7 – Finding a Mathematical Model
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Example 7 – Solution
Planets Near the Sun
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Example 7 – Solution
𝑚 =0.632 − 0
0.421≈ 1.5 =
3
2
cont’d
The equation of the line:
Y=3
2X, where Y = ln x and X = ln x
ln 𝑦 =3
2ln 𝑥