Download Section 10.3 3. a. log(1000)=3because1000=103 b. log(10

Section 10.3
1. The graph ofy = log(x) appears to be the mirror
image of the graph of y = lOxwith the line y = x
as the mirror.
3. a. log (1000)= 3 because1000= 103
b. log (10)= 1because10= 101
C. log (1) = 0 because1 = 10°
d. 10g ~
( )=
100
-2 because
~ = 10-2
100
5. 10g(l)< log(3.2) < 10g(l0)
o <log(3.2) <1
log (3.2) is between 0 and 1.
7.
6(IW' = 600
(lW' = 100
10g(lW' =log(102) Take the log on both sides
and write 100 as 102
The log undoes the
base 10 exponential
2x=2
x=1
9. a. log(600)= x
2.77815..x
b. log G)
= 1.5x
10gG)
-=x
1.5
-.31808= x
c.
log(.2)= 3x + 1
log(.2)-1 = 3x
log(.2)-1 =x
3
-.56632.. x
11. a. x = 102= 100
b. x = 103= 1000
c. x = 101= 10
148
Chapter 10: Exponential and Logarithmic Functions
13. Yl = 2.7, Y2 = log (X)
We need a large enough value for Xmax so that
log (x) > 2.7. log (1000) = 3 so we canuse the
mendly value, Xmax = 940. Given that
Yl = 2.7, we can set Ymin = 0 and Ymax = 5.
Windows may vary: Xmin = 0, Xmax = 940,
Ymin = 0, Ymax = 5
b. 250=500(~J
Yl = 250,Y2 = 500(3/4)"X
Windowsmay vary:Xmin= 0, Xmax= 4.7,
Ymin= 0, Ymax= 600
I"'-
Irahrstc:ti(lfl
M=2,'t091t201! _Y=250
The itemis worthhalfof its initialvaluein
.
Irahrstc:ti(lfl
M=501.1II723 _ Y=2.7
X"" 501.187
The answerchecksbecauselog(501.187) 2.7
""
15. You should try to follow the steps to solving this
exercise. Many teachers may prefer to leave this
exercise for future courses. See Example 8 on
page 407 for a similar problem.
.5 = .9i
From the definition of the common logarithm on
page 404 of the text, x may be written as
approximately 2.4 years.
19. a. Ao= 750,r = .038,n = 12
SubstitutingforAo,r, n in the general
compound formula, A(t) = (1 + ~
r
121
A(t)
= 750
(
1+ .~~8
)
121
10Iog(x) . Applythisto .5 and .97in the equation
to get
101og(.5)
=
( 1 01og(.97) r
(
Windows may vary: Xmin = 0, Xmax = 47,
Ymin = 0, Ymax = 2000
101og(.5) = 10/.1og(.97)
Because this is an equation, the left side must
equal the right side. Because we have the same
base on both sides of the equation, the powers
must be equal.
10g(.5)=t *log(.97)
log(.5)
-=t
10g(.97)
For an approximate solution, evaluate the
expression for t on a calculator.
t"" 22.76
Skills and Review 10.3
1 3
17. a. Yo=500, b=l--=-,k=
1
4 4
Substituting for Yo, b, k into the general
exponential function y = yobb:,we have
y=500GJ
)
b. 1500= 750 1 + .~~8
Yl = 1500,Y2= 750(1+ .038/l2)"(12X)
Irahrstc:ti(lfl
M= 111.2':;95 B1 _ Y=15 00 ..............The account is worth $1,500 in approximately
18.3 years.
21. y = a,;; find a ifx = 5 andy = 175.
175=a*(5)2
175 = 25a
7=a
23. (c+3t=16
To undo the 4thpower on the left side raise to the
power y. on both sides. Be sure to include the ::I:
when taking an even root.
1/4
(
(c+W ) =:!:(16)1/4
c+3=:!:2
c = -3:!:2
c=-3+20r-3-2
c=-10rc=-5
@ Houghton Mifflin Company. All rights reserved.
25.
2 4 X
-+-=-
3 5 30
Combine the fractions on left side, then use the
cross multiplication property to solve for x.
2*5 4*3
x
-+-=3 * 5 5 *3 30
10 12 X
-+-=15 15 30
22 x
15 30
22 *30 = 15x
660 = 15x
44=x
An alternative approach is to clear fractions by
multiplying on both sides by 30. Although this
method may save steps, it often leads to confusion
when we simplify rational expressions and solve
rational equations in Chapter II.