To prove the power rule, let
Convert to exponential form.
Raise to the power r.
Power rule for exponents
Convert to logarithmic form; commutative property
from aboveThis is the statement to be proved.
As a special case of the power rule, let so
For example, using this result, with
and
Another special case is
logb
1
x
=logb x-^1 = -logb x.
23 x^4 = logb x4/3=
4
3
25 x=logb x1/5= logb logb x.
1
5
logb logb x
x 7 0,
2
px= logb x1/p=
1
p
logb logb x.
r=^1 p ,
logb xr= r logb x m=logb x
logb xr= rm
bmr= xr
1 bm 2 r= xr
bm= x
logb x= m.
598 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions
Using the Power RuleUse the power rule to rewrite each logarithm. Assume b 7 0,x 7 0,and bZ1.
EXAMPLE 3
(a)
= 2 log 5 4 Power rule
log 5 42 (b)
= 5 logb x Power rule
logb x^5
(c)
= Power rule
1
2
logb 7
= logb 7 1/2 2 x=x1/2
logb 27 (d)
Power ruleNOW TRY=
2
5
log 2 x
= log 25 x^2 =x2/5
2 x
2/5log 225 x^2
Two special properties involving both exponential and logarithmic expressions
come directly from the fact that logarithmic and exponential functions are inverses of
each other.
Special PropertiesIf and then the following are true.
blogb^ xx, x> 0 and logb bxx
b 70 bZ 1,
To prove the first statement, let
Convert to exponential form.
Replace ywith.The proof of the second statement is similar.
blogb^ x= x logb x
by= x
y= logb x
y=logb x.
NOW TRY
EXERCISE 3
Use the power rule to rewrite
each logarithm. Assume
and
(a) (b)
(c)log 324 x^3
log 7 53 loga 210a 7 0, x 7 0, aZ1.
NOW TRY ANSWERS
- (a) (b)
(c)^34 log 3 x1
3 log 7 5 2 loga 10