8.5 Properties of Logarithms

8.5 Properties of LogarithmsGoal: Use the properties of logarithms to evaluate, expand and condense logarithmic expressions.Warm-upSimplify:

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 10-12 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 change-of-base 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