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Properties of Logarithms

- Lesson 5.5

Basic Properties of Logarithms

- Note box on page 408 of text
- Most used properties

Using the Log Function for Solutions

- Consider solving
- Previously used algebraic techniques(add to,

multiply both sides) not helpful - Consider taking the log of both sides and using

properties of logarithms

Try It Out

- Consider solution of1.7(2.1) 3x 2(4.5)x
- Steps
- Take log of both sides
- Change exponents inside log to coefficients

outside - Isolate instances of the variable
- Solve for variable

Natural Logarithms

- We have used base of 10 for logs
- Another commonly used base for logs is e
- e is an irrational number (as is )
- e has other interesting properties
- Later to be discovered in calculus
- Use ln button on your calculator

Properties of the Natural Logarithm

- Recall that y ln x ? x ey
- Note that
- ln 1 0 and ln e 1
- ln (ex) x (for all x)
- e ln x x (for x gt 0)
- As with other based logarithms

Use Properties for Solving Exponential Equations

- Given
- Take log ofboth sides
- Use exponent property
- Solve for whatwas the exponent

Misconceptions

- log (ab) NOT the same as log a log b
- log (a-b) NOT the same as log a log b
- log (a b) NOT same as (log a)(log b)
- log (a/b) NOT same as (log a)/(log b)
- log (1/a) NOT same as 1/(log a)

Usefulness of Logarithms

- Logarithms useful in measuring quantities which

vary widely - Acidity (pH) of a solution
- Sound (decibels)
- Earthquakes (Richter scale)

Chemical Acidity

- pH defined as pH -logH
- where H is hydrogen ion concentration
- measured in moles per liter
- If seawater is H 1.110-8
- then log(1.110-8) 7.96

Chemical Acidity

- What would be the hydrogen ion concentration of

vinegar with pH 3?

Logarithms and Orders of Magnitude

- Consider increase of CDs on campus since 1990
- Suppose there were 1000 on campus in 1990
- Now there are 100,000 on campus
- The log of the ratio is the change in the order

of magnitude

Decibels

- Suppose I0 is the softest sound the human ear

can hear - measured in watts/cm2
- And I is the watts/cm2 of a given sound
- Then the decibels of the sound is

The log of the ratio

Logarithms and Orders of Magnitude

- We use the log function because it counts the

number of powers of 10 - This is necessary because of the vast range of

sound intensity that the human ear can hear

Decibels

- If a sound doubles, how many units does its

decibel rating increase?

Change of Base Formula

- We have used base 10 and base e
- What about base of another number
- log 2 17 ?
- Use formula
- Think how to create a function to do this on your

calculator

Use base 10 or base e which calculator can do for

you

Assignment

- Lesson 5.5
- Page 444
- Exercises 1 85 EOO
- Assign change of base spreadsheet
- Due in 1 week.

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