9.5 properties of logarithms
DESCRIPTION
9.5 Properties of Logarithms. Laws of Logarithms. Just like the rules for exponents there are corresponding rules for logs that allow you to rewrite the log of a product , the log of a quotient , or the log of a power. Log of a Product. Logs are just exponents - PowerPoint PPT PresentationTRANSCRIPT
9.5 Properties of 9.5 Properties of LogarithmsLogarithms
2
Laws of LogarithmsLaws of Logarithms
Just like the rules for exponents there are corresponding rules for logs that allow you to rewrite the log of a product, the log of a quotient, or the log of a power.
3
Log of a ProductLog of a Product
Logs are just exponents The log of a product is the sum of the logs of
the factors: logb xy = logb x + logb y
Ex: log (25 ·125) = log 25 + log 125
4
Log of a QuotientLog of a Quotient
Logs are exponents The log of a quotient is the
difference of the logs of the factors: logb = logb x – logb y
Ex: ln ( ) = ln 125 – ln 25
x
y125
25
5
Log of a PowerLog of a Power
Logs are exponents The log of a power is the product of
the exponent and the log: logb xn = n∙logb x Ex: log 32 = 2 ∙ log 3
6
Rules for LogarithmsRules for Logarithms
These same laws can be used to turn an expression into a single log:
logb x + logb y = logb xy
logb x – logb y = logb
n∙logb x = logb xn
x
y
7
logb(xy) = logb x + logb y
Express 3logAB
C
as a sum and difference of logarithms:
3logAB
C
= log3A + log3B
ExamplesExamples
logb( ) = logb x – logb y logb xn = n logb x_______________________________
Solve: x = log330 – log310
= log33
3
30= log
10
Evaluate: 5 25 125log1
25 525 125log log
5
1125
22 log
= 2 = 13
2
7
2
– log3C
x = 1
= log3 AB
x
y
8
Sample ProblemSample Problem Express as a single logarithm:
3log7x + log7(x+1) - 2log7(x+2) 3log7x = log7x3
2log7(x+2) = log7(x+2)2
log7x3 + log7(x+1) - log7(x+2)2
log7(x3·(x+1)) - log7(x+2)2
log7(x3·(x+1)) - log7(x+2)2 =
log( )7
3
2
1
2
x x
x
b glog
( )7
3
2
1
2
x x
x
b g
To use a calculator to evaluate logarithms with other bases, you can change the base to 10 or “e” by using either of the following:
For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1:
Example: Evaluate log4 22
≈ 2.2295
b
a b
log xlog x
log a
Change of Base Formula
a
log xlog x
log a
a
ln xlog x
ln a22
224
4
loglog
log1.3424
=0.6021