-
Open-Ended Write and solve a logarithmic equation.
-
Reasoning The graphs of
y
=
2
3
x
y equals , 2 super 3 to the x end super and
y
=
3
x
+
1
y equals . 3 super x plus 1 end super intersect at approximately (1.1201, 10.2692). What is the solution of
2
3
x
=
3
x
+
1
?
2 super 3 x end super , equals . 3 super x plus 1 end super . question mark
-
Reasoning If log
12
0.5
x
=
log
143.6
,
12 super 0.5 x end super . equals log , 143.6 , comma then
12
0.5
x
12 super 0.5 x end super = ___?___.
Acoustics In Exercises 62–63, the loudness measured in decibels (dB) is defined by loudness =
10
log
I
I
0
,
10 log . fraction i , over i sub 0 end fraction . comma where I is the intensity and
I
0
=
10
−
12
i sub 0 , equals 10 minus 12
W/m
2
.
cap wslashm squared , .
- The human threshold for pain is 120 dB. Instant perforation of the eardrum occurs at 160 dB.
- Find the intensity of each sound.
- How many times as intense is the noise that will perforate an eardrum as the noise that causes pain?
- The noise level inside a convertible driving along the freeway with its top up is 70 dB. With the top down, the noise level is 95 dB.
- Find the intensity of the sound with the top up and with the top down.
- By what percent does leaving the top up reduce the intensity of the sound?
Solve each equation. If necessary, round to the nearest ten-thousandth.
-
8
x
=
444
8 to the x , equals 444
-
1
2
log
x
+
log
4
=
2
1 half log x plus log 4 equals 2
-
4
log
3
2
−
2
log
3
x
=
1
4 , log base 3 , 2 minus 2 , log base 3 , x equals 1
-
log
x
2
=
2
log , x squared , equals 2
-
9
2
x
=
42
9 super 2 x end super , equals 42
-
log
8
(
2
x
–
1
)
=
1
3
log base 8 , open , 2 to the x , en dash 1 close equals , 1 third
-
log
(
5
x
−
4
)
=
3
log open 5 x minus 4 close equals 3
-
12
4
−
x
=
20
12 super 4 minus x end super . equals 20
-
5
3
x
=
125
5 super 3 x end super , equals 125
-
log
4
+
2
log
x
=
6
log 4 plus 2 log x equals 6
-
4
3
x
=
77.2
4 super 3 x end super , equals , 77.2
-
log
7
3
x
=
3
log base 7 , 3 x equals 3
Use the properties of exponential and logarithmic functions to solve each system. Check your answers.
-
{
y
=
2
x
+
4
y
−
4
x
−
1
=
0
left brace . table with 2 rows and 1 column , row1 column 1 , y equals . 2 super x plus 4 end super , row2 column 1 , y minus . 4 super x minus 1 end super . equals 0 , end table
-
{
2
x
+
y
=
16
4
x
−
y
=
1
left brace . table with 2 rows and 1 column , row1 column 1 , 2 super x plus y end super . equals 16 , row2 column 1 , 4 super x minus y end super . equals 1 , end table
-
{
log
(
2
x
−
y
)
=
1
log
(
x
+
y
)
=
3
log
2
left brace . table with 2 rows and 1 column , row1 column 1 , log open 2 x minus y close equals 1 , row2 column 1 , log open x plus y close equals 3 log 2 , end table
C Challenge
Solve each equation.
-
log
7
(
2
x
−
3
)
2
=
2
log base 7 . open 2 x minus 3 close squared . equals 2
-
log
2
(
x
2
+
2
x
)
=
3
log base 2 , open , x squared , plus , 2 to the x , close equals 3
-
3
2
log
2
4
−
1
2
log
2
x
=
3
3 halves . log base 2 , 4 minus , 1 half . log base 2 , x equals 3
-
Meteorology In the formula
P
=
P
0
(
1
2
)
h
4795
,
p equals , p sub 0 . open , 1 half , close super h over 4795 end super . comma P is the atmospheric pressure in millimeters of mercury at elevation h meters above sea level.
P
0
p sub 0 is the atmospheric pressure at sea level. If
P
0
p sub 0 equals 760 mm, at what elevation is the pressure 42 mm?
