5.0Properties of Logarithms

AB Review for Ch.5

Rules of LogarithmsIf M and N are positive real numbers and b is ≠ 1:

• The Product Rule:• logbMN = logbM + logbN

(The logarithm of a product is the sum of the logarithms)

• Example: log4(7 • 9) = log47 + log49• Example: log (10x) = log10 + log x

Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1:

• The Product Rule:• logbMN = logbM + logbN

(The logarithm of a product is the sum of the logarithms)

• Example: log4(7 • 9) = log47 + log49• Example: log (10x) = log10 + log x• You do: log8(13 • 9) =

• You do: log7(1000x) =

log813 + log89log71000 + log7x

Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1:

• The Quotient Rule

(The logarithm of a quotient is the difference of the logs)

• Example:

log log logb b bM M NN

log log log 22x x

Rules of LogarithmsIf M and N are positive real numbers and b ≠ 1:

• The Quotient Rule

(The logarithm of a quotient is the difference of the logs)

• Example:

• You do:

log log logb b bM M NN

log log log 22x x

714logx

7 7log 14 log x

Rules of LogarithmsIf M and N are positive real numbers, b ≠ 1, and p is any real

number:

• The Power Rule:• logbMp = p logbM

(The log of a number with an exponent is the product of the exponent and the log of that number)

• Example: log x2 = 2 log x• Example: ln 74 = 4 ln 7• You do: log359 =

ln x

9log35

Simplifying (using Properties)

• log94 + log96 = log9(4 • 6) = log924• log 146 = 6log 14• a

• You try: log1636 - log1612 =

• You try: log316 + log24 = • You try: log 45 - 2 log 3 =

log163

Impossible!

log 5

3log3 log 2 log2

Using Properties to Expand Logarithmic Expressions

• Expand:

Use exponential notation

Use the product rule

Use the power rule

2

12 2

12 2

log

log

log log12log log2

b

b

b b

b b

x y

x y

x y

x y

Expand: 3

6 4log36xy

6 61 log 2 4log3

x y

Condense: log log 3logb b bM N P

3logbMNP

Change of Base

• Examine the following problems:• log464 = x

» we know that x = 3 because 43 = 64, and the base of this logarithm is 4

• log 100 = x– If no base is written, it is assumed to be base 10

» We know that x = 2 because 102 = 100

• But because calculators are written in base 10, we must change the base to base 10 in order to use them.

Change of Base Formula

• Example loglog558 =8 =

• This is also how you graph in another base. Enter y1=log(8)/log(5). Remember, you don’t have to enter the base when you’re in base 10!

loglog

.8512900

blogMlog M logb

Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.

13

6logf x xy

x4

–4

: 0,Domain

: 0VA x

int : 1,0x

Graphing logarithmic functions

Find the domain, vertical asymptotes, and x-intercept. Sketch a graph.

10log 1 4g x x y

x4

–4

: 1,Domain

: 1VA x

4int : 1 10 ,0x

Graphing logarithmic functions.

Homework:MMM pg. 186-188