(d)
Power ruleQuotient rule(e)
Power ruleProduct and quotient rulesMultiply in the numerator.(f ) cannot be rewritten using the properties of logarithms. There is
no property of logarithms to rewrite the logarithm of a sum.
log 8 12 p+ 3 r 2
=logb
2 x^2 + 3 x+ 1
x2/3
=logb
1 x+ 1212 x+ 12
x2/3
=logb 1 x+ 12 +logb 12 x+ 12 - logb x2/3
logb 1 x+ 12 + logb 12 x+ 12 -
2
3
logb x, bZ 1
=logb
m^4
n
=logb m^4 - logb n
4 logb m- logb n, bZ 1
600 CHAPTER 10 Inverse, Exponential, and Logarithmic Functions
NOW TRYIn the next example, we use numerical values for and While we use
the equals symbol to give these values, they are actually just approximations since
most logarithms of this type are irrational numbers. We use with the understand-
ing that the values are correct to four decimal places.
log 2 5 log 2 3.
Using the Properties of Logarithms with Numerical ValuesGiven that and evaluate the following.
(a)
Factor 15.Product ruleSubstitute the given values.Add.(b)
Quotient ruleSubstitute the given values.Subtract.(c)
Write 27 as a power of 3.Power ruleSubstitute the given value.= 4.7550 Multiply. NOW TRY
= 31 1.5850 2
= 3 log 2 3
= log 2 33
log 2 27
=-0.7369
= 1.5850-2.3219
= log 2 3 - log 2 5
= log 2 0.6= 106 =^35
3
5
log 2 0.6
= 3.9069
= 1.5850+2.3219
= log 2 3 +log 2 5
= log 213 # 52
log 2 15
log 2 5 =2.3219 log 2 3 = 1.5850,
EXAMPLE 6
NOW TRY
EXERCISE 5
Use properties of logarithms
to rewrite each expression if
possible. Assume that all vari-
ables represent positive real
numbers.
(a)
(b)
(c)
(d)
(e) log 7 149 + 2 x 2
-
3
5
log 5 x, x 710
+log 5 1 x- 102log 5 1 x+ 102log 2 x+3 log 2 y-log 2 zlog 6
Bn
3 mlog 3 9 z^4NOW TRY ANSWERS
- (a)
(b)(c) (d)
(e)cannot be rewrittenlog 5 x(^2) - 100
x3/5
log 2
xy^3
z
1
2 1 log 6 n-log 6 3 - log 6 m^2
2 +4 log 3 z
NOW TRY
EXERCISE 6
Given that
and evalu-
ate the following.
(a) (b)
(c) log 2 49
log 2 70 log 2 0.7
log 2 10 =3.3219,
log 2 7 =2.8074
- (a)6.1293 (b)
(c)5.6148- 0.5145