Properties of LogarithmsIf x, y, and bare positive real numbers, where and ris any real number,
then the following are true.
Product Rule
Quotient Rule
Power Rule
Special Properties blogb^ xx and logb bxx
logb xrr logb x
logb
x
y
logb xlogb y
logb xylogb xlogb y
bZ1,
Using the Special PropertiesFind each value.
(a) , since (b)
(c) 4 log^4 10 = 10 NOW TRY
log 5 54 = 4 logb bx=x. log 3 9 = log 3 32 = 2
EXAMPLE 4
SECTION 10.4 Properties of Logarithms 599
OBJECTIVE 4 Use properties to write alternative forms of logarithmic
expressions.
Writing Logarithms in Alternative FormsUse the properties of logarithms to rewrite each expression if possible. Assume that
all variables represent positive real numbers.
(a)
Product rule; power rule(b)
Write the radical expression
with a rational exponent.Power ruleQuotient rule(c)
Quotient rulePower ruleProduct rule=2 log 5 a-log 5 b-log 5 c
=2 log 5 a- 1 log 5 b+log 5 c 2
=2 log 5 a-log 5 bc
=log 5 a^2 - log 5 bc
log 5
a^2
bc
=
1
2
1 log 7 m-log 7 n 2
=
1
2
log 7
m
n
=log 7 a
m
n
b
1/2log 7
B
m
n
= 1 +3 log 4 x log 4 4 = 1
=log 4 4 +log 4 x^3
log 4 4 x^3
EXAMPLE 5
Parentheses are
necessary here.We summarize the properties of logarithms.
NOW TRY
EXERCISE 4
Find each value.
(a)
(b)
(c) 8 log^8 5
log 10 10,000log 4 47NOW TRY ANSWERS
- (a) 7 (b) 4 (c) 5