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3.3 Properties of Logarithms
Objective 1: Use the product rule.
Use the product rule to expand each logarithmic expression.
log 3 (9 ⋅ 5)
log (1000x)
Objective 2: Use the quotient rule.
Use the quotient rule to expand each logarithmic expression.
⎛ 25 ⎞
log 5 ⎜ ⎟
⎝ x ⎠
⎛ x⎞
log ⎜ ⎟
⎝8⎠
⎛ e3 ⎞
ln ⎜ ⎟
⎝7⎠
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Objective 3: Use the power rule.
Use the power rule to expand each logarithmic expression.
log 5 7 2
logg 2 ((8 x) 4
log x
ln(6e)5
Objective 4: Expand logarithmic expressions.
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Study Tip
Use properties of logarithms to expand each logarithmic expression as much as possible. ⎛ x⎞
log 5 ⎜⎜
⎟⎟
⎝ 25 ⎠
log 5
x
y
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xy 4
log
16
5
⎡ x 4 x2 + 3 ⎤
ln ⎢
⎥
5
⎢⎣ ( x + 3) ⎥⎦
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Objective 5: Condense logarithmic expressions.
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. log 250 + log 4
log3 405 – log3 5
log (3x + 7) ‐ logx
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1/3 ln x + ln y
8 ln (x + 9) – 4 ln x
1/3[5 ln(x + 6) – ln x – ln(x2 – 25)]
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log x + log(x2 – 4) – log 15 – log (x + 2)
Objective 6: Use the change‐of‐base property.
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Graphing Calculator
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.
log6 17
log16 57.2
log0.3 19
logπ 400
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Use a graphing calculator and the change‐of‐base property to graph each function.
y = log15 x
y = log3 (x – 2)
9