11.4Properties of Logarithms

LogarithmsA logarithm is an operation, a little like taking

the sine of an angle.

Raising a constant to a power is called exponentiation.

There has to be an “undo” button for exponentiation.

Logarithms “undo” exponentials, because a logarithm is the inverse operation to exponentiation.

Notation & FormsMy notation:

The book’s notation:€

logb n = x ⇔ bx = n

€

loga x = y ⇔ ay = x

Properties of LogarithmsThe set-up:

Suppose that m and n are positive numbers and b is a positive number other than 1.

Let p be any real number.

The following properties hold:

Product Property

€

logb m ⋅n = logb m + logb n

€

why ?

€

Example :

Write the following as a sum or difference only :

log3 x( ) ⋅ x +1( )[ ]

Quotient Property

€

logbm

n= logb m − logb n

€

Example :

Write the following as a sum or difference only :

log5

x

x + 2

Power Property

€

logb mp = p ⋅logb m

€

Example :

Write the following as a sum or difference only :

log14

2x

x + 7 ⎛ ⎝

⎞ ⎠

3

Stop Here

Putting it all togetherWrite the following as a sum or difference only:

€

log6

3x ⋅ y − 4( )5x + 7( )

⎛

⎝ ⎜ ⎞

⎠ ⎟

8

Change of BaseYour calculator is NOT smart!

Your calculator only does logs in base 10 or base e.

What if you had to compute a log in another base? (happens all the time)

Change of base formula:

€

logb n =log10 n

log10 b=

logn

logb

When Would I Use a Logarithm?

We use logarithms to solve for variable exponents.

Example: Solve for x.

€

3x+4 = 81

ExampleSolve for x.

€

logx 56 = 2.5

ClassworkPg. 626 # 19-39 [2 each section], 40-46 [3]

HomeworkPg. 639 # 17, 19, 21, 23, 25-30