B Apply
-
Think About a Plan The pH of a substance equals
−
log
[
H
+
]
,
negative log left bracket , cap h to the plus , right bracket comma where
[
H
+
]
left bracket , cap h to the plus , right bracket is the concentration of hydrogen ions, and it ranges from 0 to 14. A pH level of 7 is neutral. A level greater than 7 is basic, and a level less than 7 is acidic. The table shows the hydrogen ion concentration
[
H
+
]
left bracket , cap h to the plus , right bracket for selected foods. Is each food basic or acidic?
Approximate
[
H
+
]
left bracket , cap h to the plus , right bracket of Foods
| Food |
[
H
+
]
left bracket , cap h to the plus , right bracket
|
| Apple juice |
3.2
×
10
−
4
3.2 times , 10 super negative 4 end super
|
| Buttermilk |
2.5
×
10
−
5
2.5 times , 10 super negative 5 end super
|
| Cream |
2.5
×
10
−
7
2.5 times , 10 super negative 7 end super
|
| Ketchup |
1.3
×
10
−
4
1.3 times , 10 super negative 4 end super
|
| Shrimp sauce |
7
.
9
×
10
−
8
. 9 times , 10 super negative 8 end super
|
| Strained peas |
1.0
×
10
−
6
1.0 times , 10 super negative 6 end super
|
- How can you find the pH value of each food?
- What rule can you use to determine if the food is basic or acidic?
-
Chemistry Find the concentration of hydrogen ions in seawater, if the pH level of seawater is 8.5.
Write each equation in exponential form.
-
log
2
128
=
7
log base 2 , 128 equals 7
-
log
0.0001
=
−
4
log , 0.0001 , equals negative 4
-
log
6
6
=
1
log base 6 , 6 equals 1
-
log
4
1
=
0
log base 4 , 1 equals 0
-
log
7
16
,
807
=
5
log base 7 , 16 comma 807 equals 5
-
log
2
1
2
=
−
1
log base 2 . 1 half , equals negative 1
-
log
3
1
9
=
−
2
log base 3 . 1 ninth , equals negative 2
-
log
10
=
1
log , 10 equals 1
Find the greatest integer that is less than the value of the logarithm. Use your calculator to check your answers.
- log 5
- log 0.08
- log 17.52
-
log
(
1.3
×
10
7
)
log open 1.3 times , 10 to the seventh , close w
-
Error Analysis Find the error in the following evaluation of
log
27
3
.
log base 27 , 3 . Then evaluate the logarithm correctly.
-
Writing Explain why the base b in
y
=
y equals
log
b
x
log base b , x cannot equal 1.
-
Open-Ended Write a logarithmic function of the form
y
=
y equals
log
b
x
.
log base b , x . Find its inverse function. Graph both functions on one set of axes.
Image Long Description
Find the inverse of each function.
-
y
=
log
4
x
y equals , log base 4 , x
-
y
=
log
0.5
x
y equals , log base 0.5 , x
-
y
=
log
10
x
y equals , log base 10 , x
-
y
=
log
2
2
x
y equals , log base 2 . 2 to the x
-
y
=
log
(
x
+
1
)
y equals log open x plus 1 close
-
y
=
log
10
x
y equals log , 10 to the x
-
y
=
log
2
4
x
y equals , log base 2 . 4 to the x
-
y
=
log
(
x
−
6
)
y equals log open x minus 6 close
Graph each logarithmic function.
-
y
=
log
2
x
y equals log , 2 to the x
-
y
=
2
log
2
x
y equals 2 , log base 2 , x
-
y
=
log
4
(
2
x
+
3
)
y equals , log base 4 , open 2 x plus 3 close
-
y
=
log
3
(
x
+
5
)
y equals , log base 3 , open x plus 5 close
Find the domain and the range of each function.
-
y
=
log
5
x
y equals , log base 5 , x
-
y
=
3
log
x
y equals 3 log x
-
y
=
log
2
(
x
−
3
)
y equals , log base 2 , open x minus 3 close
-
y
=
2
log
(
x
−
2
)
y equals 2 log open x minus 2 close
You can write
5
3
=
125
5 cubed , equals 125 in logarithmic form using the fact that
log
b
b
x
=
x
.
log base b . b to the x , equals x .
log
5
(
5
3
)
=
log
5
(
125
)
Apply the log base 5 to each side.
3
=
log
5
125
Use
log
b
b
x
=
x
to simplify.
table with 2 rows and 4 columns , row1 column 1 , log base 5 , open , 5 cubed , close , column 2 equals , column 3 log base 5 , open 125 close , column 4 cap apply the log base 5 to each side. , row2 column 1 , 3 , column 2 equals , column 3 log base 5 , 125 , column 4 cap use . log base b . b to the x , equals x . to simplify. , end table
Use this method to write each equation in logarithmic form. Show your work.
-
3
4
=
81
3 to the fourth , equals 81
-
x
4
=
y
x to the fourth , equals y
-
6
8
=
a
+
1
6 to the eighth , equals eh plus 1
