Download Section 3.3: Properties of Logarithms Study Guide Complete the

SECTION 3.3: PROPERTIES OF LOGARITHMS
STUDY GUIDE
RECALL:
LOGARITHMS
ARE EXPONENTS
COMPLETE THE TABLE:
LAW OF EXPONENTS REVIEW
PRODUCT RULE
QUOTIENT RULE
a m a n
POWER RULE
a m n
a m n
THE PRODUCT RULE
EXAMPLE
LET b, M, AND N BE POSITIVE REAL NUMBERS WITH b  1.
USE THE PRODUCT RULE TO
EXPAND
__________________________________________
LN (4X)
THE LOGARITHM OF A PRODUCT IS THE
___________________
__________________________________________
THE QUOTIENT RULE
LET b, M, AND N BE POSITIVE REAL NUMBERS WITH b  1.
EXAMPLE
__________________________________________
USE THE QUOTIENT RULE TO
EXPAND log 3 5
7
THE LOGARITHM OF A QUOTIENT IS THE
___________________

___________________________________________
THE POWER RULE
EXAMPLE
LET b, M, AND N BE POSITIVE REAL NUMBERS WITH b  1,
AND LET p BE ANY REAL NUMBER.
USE THE POWER RULE TO EXPAND
5
__________________________________________
THE LOGARITHM OF A NUMBER WITH AN EXPONENT IS THE
__________________________________________
__________________________________________
log 4
___________________
TO EXPAND A LOGARITHMIC EXPRESSION
WHEN USING
THE PRODUCT RULE
WHEN USING
QUOTIENT RULE
WHEN USING
POWER RULE
WRITE A SINGLE LOGARITHM AS
WRITE A SINGLE LOGARITHM AS
THE SUM OF TWO LOGARITHMS
THE _____________________
“PULL THE ________________
OF TWO LOGARITHMS
TO THE FRONT”
NOTE: WHEN EXPANDING A LOGARITHMIC EXPRESSION, DETERMINE WHETHER THE
REWRITING HAS CHANGED THE DOMAIN OF THE EXPRESSION.
FOR EXAMPLE:
USE A GRAPHING UTILITY TO GRAPH ln x 2 AND 2 ln x
DOMAIN: 2 ln x
DOMAIN: ln x 2
 ln x 2 = 2 ln x
IF AND ONLY IF (IFF)
__________________
CHECKPOINT 3.3.1
USE THE PRODUCT RULE TO EXPAND EACH LOGARITHMIC EXPRESSION:
log 6 7  11
log 100x 
CHECKPOINT 3.3.2
USE THE QUOTIENT RULE TO EXPAND EACH LOGARITHMIC EXPRESSION:
log 8
23x 
5
ln  e1 1 
 
CHECKPOINT 3.3.3
USE THE POWER RULE TO EXPAND EACH LOGARITHMIC EXPRESSION:
log 6 3 9
ln 3 x
CHECKPOINT 3.3.4
USE LOGARITHMIC PROPERTIES TO EXPAND EACH LOGARITHMIC EXPRESSION AS MUCH AS POSSIBLE:

log b x 4 3 y

 x
log 5 
 25y 3





REMEMBER:
EQUALS MEANS “SAME”
TO CONDENSE A LOGARITHMIC EXPRESSION
WRITE THE SUM OR DIFFERENCE OF TWO OR MORE LOGARITHMIC EXPRESSIONS
AS A ____________________________________________.
1. USE THE PROPERTIES OF LOGARITHMS.
2. COEFFICIENTS OF LOGARITHMS MUST BE ______ BEFORE YOU CAN CONDENSE
THEM USING THE PRODUCT AND ____________________ RULES.
 FOR EXAMPLE, TO CONDENSE 2 ln x  lnx  1 THE COEFFICIENT OF THE
FIRST TERM MUST BE _____. USE THE _____________________ TO
REWRITE THE COEFFICIENT AS AN EXPONENT.


 2 ln x  lnx  1  ln x 2  lnx  1  ln x 2 x  1
CHECKPOINT 3.3.5
WRITE AS A SINGLE LOGARITHM
log 25  log 4
log7x  6   log x
CHECKPOINT 3.3.6
WRITE AS A SINGLE LOGARITHM
2 logx  3   log x
ln x  31 lnx  5 
1
4
log b x  2 log b 5  10 log b y
THE CHANGE OF BASE PROPERTY
FOR ANY LOGARITHMIC BASES a AND b,
AND ANY POSITIVE NUMBER M,
__________________________________________
THE LOGARITHM OF M WITH BASE B IS EQUAL TO
THE LOGARITHM OF M WITH ANY NEW BASE DIVIDED BY
THE LOGARITHM OF B WITH THAT NEW BASE.
CHANGE OF BASE PROPERTY
USING THE COMMON LOGARITHM
CHANGE OF BASE PROPERTY
USING THE NATURAL LOGARITHM
_______________________
_______________________
CHECKPOINT 3.3.7
USE COMMON LOGARITHMS TO EVALUATE
CHECKPOINT 3.3.8
USE NATURAL LOGARITHMS TO EVALUATE
log 7 2506
log 7 2506