Exponents and Logarithms
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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
3
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
6
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
7
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
8
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