8.5 Properties of Logarithms Goal: Use the properties of logarithms to evaluate, expand and condense...

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8.5 Properties of Logarithms
8.5 Properties of LogarithmsGoal: Use the properties of logarithms to evaluate, expand and condense logarithmic expressions.WarmupSimplify:
Properties of LogarithmsProperties of LogarithmsLet b, m, and n be positive numbers such that b 1.Product Property
Quotient Property
Power Property
Example 1
Example 2Expand the expression. Assume that all variables are positive.
Example 3Condense the expression.
Loudness of a SoundDecibel LevelExample130Jackhammer110Dance club90Lawn mower70Television50Rainfall30Soft whisper10Rustling leafThe loudness L of a sound (in decibels) is related to the intensity I of the sound (in watts per square meter) by the equation
where I0 is an intensity of 1012 watts per square meter, roughly the faintest sound that can be heard by humans.An air horn emits sound with an intensity I of 1 watt per square meter. Find its decibel level.
Loudness of a Sound
How much louder is the sound of 5 air horns, compared to just one air horn?
So 5 air horns would be approximately 7 decibels more.
Change of Base FormulaLogarithms with any base other than 10 or e can be written in terms of common or natural logarithms using the changeofbase formula. This allows you to evaluate any logarithm using a calculator.Change of Base Formula
If a, b, and c are positive numbers with b 1 and c 1, then:
In particular, and
ExampleEvaluate:
AssignmentWorksheet 8.5Blue AgeSoundzaboundVolume 1, track 141998New Age98300.555