5.3 Properties of Logarithms

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5.3 Properties of Logarithms

• Use the change of base formula to rewrite and
  evaluate logs
• Use properties of logs to evaluate or rewrite log
  expressions
• Use properties of logarithms to expand or
  condense logarithmic expressions
• Use logarithmic functions to model and solve
  real-life problems.

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Change of Base Formula

• Base b Base e
• logax logbx logax ln x
• logba ln a

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Using the Change of Base Formula

• Examples
• log4 25

Rewrite as log 25 log 4
1.39794 .060206
2.3219
Rewrite as log 12 log 2
1.07918 .30103
log2 12
3.5850
The same 2 problems can be done using ln.
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Properties of Logarithms

• Product Property
• loga (uv) loga u loga v
• Quotient Property
• loga (u/v) loga u - loga v
• Power Property
• loga un n loga u

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Using Properties of Logs to find the exact value
of the expression

• Example
• log5 3?5

ln e6 ln e2
Bring exponents out front. 6ln e 2ln e
Rewrite-- log5 (5)1/3
So-- 6 2 4
Bring exponent out front. 1/3 log5 (5)
1/3
OR we could have rewritten this as division
Ln e6 lne4 4lne 4 e2
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Using Properties of Logarithms to expand the
expression as a sum, difference and/or constant

• ln 2/27

ln 2 - ln 27
log310z

• log310 log3z

log 4x2y

• log 4 log x2 log y

• log 4 2log x log y

ln 6 ?x2 1
ln 6 ln (x2 1)1/2
ln 6 1/2ln (x2 1)
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Write the expression as a single logarithm (Go
Backwards)

• ln y ln t

log 8 log t
log 8/t
ln yt
-4ln 2xt
ln (2xt)-4
2 ln 8 5 ln (x 4)
ln 82 ln (x 4)5
ln 82(x 4)5
1/3log x log (x 1)
log x(x 1)1/3
23ln x ln (x 1) ln(x 1)
3ln x ln (x 1) ln(x 1)2
ln x3 2 (x 1)(x
1)
Foil this