7-3 Logarithmic Functions as Inverses
Quick Review
If
x
=
b
y
,
x equals , b to the y , comma then
log
b
x
=
y
.
log base b , x equals y . The logarithmic function is the inverse of the exponential function, so the graphs of the functions are reflections of one another across the line
y
=
x
.
y equals x . Logarithmic functions can be translated, stretched, compressed, and reflected, as represented by
y
=
a
log
b
(
x
−
h
)
+
k
,
y equals eh , log base b , open x minus h close plus k comma similarly to exponential functions. When
b
=
10
,
b equals 10 comma the logarithm is called a common logarithm, which you can write as log x.
Example
Write
5
−
2
=
0.04
5 super negative 2 end super , equals , 0.04 in logarithmic form.
If
y
=
b
x
,
y equals , b to the x , comma then
log
b
y
=
x
.
log base b , y equals x .
y
=
0.
y equals 0. 04,
b
=
5
b equals 5 and
x
=
−
2
.
x equals negative 2 .
So,
log
5
0.04
=
−
2
.
log base 5 . 0.04 , equals negative 2 .
Exercises
Write each equation in logarithmic form.
-
6
2
=
36
6 squared , equals 36
-
2
−
3
=
0.125
2 super negative 3 end super , equals , 0.125
-
3
3
=
27
3 cubed , equals 27
-
10
−
3
=
0.001
10 super negative 3 end super , equals , 0.001
Evaluate each logarithm.
-
log
2
64
log base 2 , 64
-
log
3
1
9
log base 3 . 1 ninth
- log 0.00001
-
log
2
1
log base 2 , 1
Graph each logarithmic function.
-
y
=
log
3
x
y equals , log base 3 , x
-
y
=
log
x
+
2
y equals log x plus 2
-
y
=
3
log
2
(
x
)
y equals 3 , log base 2 , open x close
-
y
=
log
5
(
x
+
1
)
y equals , log base 5 , open x plus 1 close
How does the graph of each function compare to the graph of the parent function?
-
y
=
3
log
4
(
x
+
1
)
y equals 3 , log base 4 , open x plus 1 close
-
y
=
−
ln
x
+
2
y equals negative ln x plus 2
7-4 Properties of Logarithms
Quick Review
For any positive numbers, m, n, and b where
b
≠
1
,
b not equal to 1 comma each of the following statements is true. Each can be used to rewrite a logarithmic expression.
-
log
b
log base b
m
n
=
m n equals
log
b
m
+
log
b
n
,
log base b , m plus , log base b , n comma by the Product Property
-
log
b
m
n
=
log
b
m
−
log
b
n
,
log base b . m over n , equals , log base b , m minus , log base b , n . comma by the Quotient Property
-
log
b
m
n
=
n
log
b
m
,
log base b . m to the n , equals n , log base b , m comma by the Power Property
Example
Write
2
log
2
y
+
log
2
x
2 , log base 2 , y plus , log base 2 , x as a single logarithm. Identify any properties used.
2
log
2
y
+
log
2
x
2 , log base 2 , y plus , log base 2 , x
=
log
2
y
2
+
log
2
x
Power Property
=
log
2
x
y
2
Product Property
table with 2 rows and 2 columns , row1 column 1 , equals , log base 2 . y squared , plus , log base 2 , x , column 2 cap power cap property , row2 column 1 , equals , log base 2 . x y squared , column 2 cap product cap property , end table
Exercises
Write each expression as a single logarithm. Identify any properties used.
-
log
8
+
log
3
log 8 plus log 3
-
log
2
5
−
log
2
3
log base 2 , 5 minus , log base 2 , 3
-
4
log
3
x
+
log
3
7
4 , log base 3 , x plus , log base 3 , 7
-
log
x
−
log
y
log x minus log y
-
log
5
−
2
log
x
log 5 minus 2 log x
-
3
log
4
x
+
2
log
4
x
3 , log base 4 , x plus 2 , log base 4 , x
Expand each logarithm. State the properties of logarithms used.
-
log
4
x
2
y
3
log base 4 . x squared , y cubed
-
log
4
s
4
t
log . 4 s to the fourth , t
-
log
3
2
3
log base 3 . 2 thirds
-
log
(
x
+
3
)
2
log . open x plus 3 close squared
-
log
2
(
2
y
−
4
)
3
log base 2 , open 2 y minus 4 close 3
-
log
z
2
5
log . fraction z squared , over 5 end fraction
Use the Change of Base Formula to evaluate each expression.
-
log
2
7
log base 2 , 7
-
log
3
10
log base 3 , 10
