Transcript Exponents and Logarithms

4.3 Laws 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.
2
Log 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
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Log of a Quotient
 Logs are exponents
 The log of a quotient is the
difference of the logs of the factors:
x
 logb
y
= logb x – logb y
 Ex: ln ( 125 ) = ln 125 – ln 25
25
4
Log 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
5
Rules for Logarithms
 These same laws can be used to turn an
expression into a single log:
 logb x + logb y = logb xy
x
 logb x – logb y = logb
y
 n∙logb x = logb xn
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Examples
logb(xy) = logb x + logb y
logb( x ) = logb x – logb y
logb xn = n logb x
_______________________________
y

 as a sum and difference of logarithms:
Express log3 

 C 
AB
 AB 
log3 
= log=3Alog
+ 3log
AB3B – log3C

 C 
Solve: x = log330 – log310
 30 
= log 3  
 10 
= log33
x=1
Evaluate: log5 25 125
 log5 25  log5 125
1
 2  log 5 125
2
1
7
= 2   3 =
2
2
1
2
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Sample 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 =
b g
x3 x  1
log 7
( x  2) 2
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Change of Base Formula
For all positive numbers a, b, and x, where a ≠ 1 and b ≠ 1:
logb x
loga x 
logb a
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:
log x
loga x 
log a
Example:
Evaluate log4 22
ln x
loga x 
ln a
log 22
log4 22 
log 4
1.3424
=
0.6021
≈ 2.2295