5.0 Properties of Logarithms AB Review for Ch.5. Rules of Logarithms If M and N are positive real...
-
Upload
arnold-greene -
Category
Documents
-
view
227 -
download
0
description
Transcript of 5.0 Properties of Logarithms AB Review for Ch.5. Rules of Logarithms If M and N are positive real...
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