Transcript Angles, Degrees, and Special Triangles

Properties of Logarithms
MATH 109 - Precalculus
S. Rook
Overview
• Section 3.3 in the textbook:
– Properties of logarithms
– Change-of-base formula
– Logarithmic scales
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Properties of Logarithms
Properties of Logarithms
• Logarithms can be manipulated using a set of very
important properties:
– Product: loga(uv) = logau + logav
• NOTE: loga u  loga v  loga u  v  loga u  loga v
– Quotient: loga(u⁄v) = logau – logav
• NOTE: loga u  log u  v   log u  log v
loga v
a
a
a
– Power: loga(un) = n ∙ logau
• Applicable to logarithms with ANY valid base including
common and natural logarithms
• The bases of the logarithms MUST be the same
• Used to write equivalent logarithmic expressions
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Expanding & Compressing Logarithms
• Tips when expanding one logarithm into multiple
logarithms with the SAME base as the original:
– Work from outer to inner
• Tips when compressing several logarithms of the SAME
base into one logarithm of that SAME base:
– Apply the power property if necessary
• Removes coefficients from in front of logarithms
• Logarithms must NOT have a coefficient in front when
combining
– Work from inner to outer
– Apply the product and quotient properties of
logarithms to combine
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Expanding Logarithms (Example)
Ex 1: Use the properties of logarithms to
expand the expression as a sum, difference,
and/or constant multiple of logarithms:
a) ln xyz
2
x4 y
b) log5 z 3
c) log 4 x 3 x 2  3
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Compressing Logarithms (Example)
Ex 2: Condense the expression to the logarithm
of a single quantity:
a) ln x  3 lnx  1
b) log x  2 log y  3 log z


1
c) 2 lnx  3  ln x   ln x 2  1
3
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Change-of-Base Formula
Change-of-Base Formula
• Recall last lesson when we discussed that the
calculator can only evaluate in base 10 (log) or
base e (ln)
– Also mentioned that we could “trick” the calculator into
evaluating in other bases
• Change-of-Base Formula:
log x ln x
logb x 

log b ln b
– Note that the base in the ratios can be any value – just
as long as it is the SAME base
• e.g.
log32 ln 32
log2 32 

5
log 2
ln 2
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Change-of-Base Formula (Example)
Ex 3: Approximate the logarithm to three
decimal places using the change-of-base
formula with a) log b) ln:
2
a) log 9
5
b) log15 1250
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Logarithmic Scales
Logarithmic Scales
• Used to scale very large or very small numbers
to a more easily understood interval
• We will see this applied with the Richter Scale
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Richter Scale Magnitude
• The Richter scale is used to convert earthquake
intensities to a 0 to 10 scale
– A logarithmic scale is required because the intensities can
grow extremely large
• Because intensities are scaled down so compactly,
the difference in intensities between any two
numbers on the 0 to 10 scale is significant
• Richter Scale Magnitude:
 I 
M  log  where M is themagnitude,I is the
 I0 
intensity,and I 0 is theintensityof a zero levelearthquake13
Richter Scale Magnitude (Example)
Ex 4: Compare the intensity of an earthquake
that measured 4.5 on the Richter Scale with
an earthquake that measured 5.5 on the
Richter Scale
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Summary
• After studying these slides, you should be able to:
– Use the properties of logarithms to condense and expand
logarithmic expressions
– Apply the change-of-base formula for bases other than e
or 10
– Solve application problems involving logarithmic scales
• Additional Practice
– See the list of suggested problems for 3.3
• Next lesson
– Exponential & Logarithmic Equations (Section 3.4)
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