Math 121. Logarithmic Functions
Name:
1
) in exponential form.
2. (a) Write y = log5 ( 125
1. (a) Write y = logb x in exponetial form.
(b) Write y = log3 81 in exponetial form.
Answer: x = by
1
(c) Evaluate log2 ( 16
).
(b) Use the exponential form of y = log2 x, that
is x = 2y to complete the following table.
1
8
1
4
1
2
0
2
4
8
y
−3
−2
−1
0
1
2
3
3
1
125
(b) 2y =
1
(c) y = log2 ( 16
) implies 2y =
y = −4
(b) Use the exponential form of y = log 1 x, that
is x =
(d) Evaluate log 1 243
Answer: (a) 5y =
x = 2y
( 12 )y
Hints and Answers
2
to complete the following table.
x = ( 12 )y
8
4
2
1
1
2
1
4
1
8
y
−3
−2
−1
0
1
2
3
1
16
1
32
= 2−4 and so
(d) y = log 1 243 implies ( 13 )y = 243 and so
3
3−y = 35 and y = −5.
3. A graph of y = bx is given below. On the
same axes, graph
(a) f (x) = logb x
(b) g(x) = logb (x + 2) + 3
(c) Sketch f (x) = log2 x and g(x) = log 1 x on
2
the axes below
(c) h(x) = log 1 x
b
(d) How are the graphs of f and g related?
Answer: They are reflections over the x-axis.
(e) In general, for b > 1 how are the graphs of
y = logb x and y = log 1 x related?
b
Answer: They are reflections over the x-axis.
4. (a) Find the domain for f (x) = log7 (x2 − 4).
(b) Find the domain of g(x) = log5 (−x).
(c) Find the domain of h(x) = log3 (5 − x).
(f) Given f (x) = logb x for b > 0 and b 6= 1
The domain of f is:
The range of f is:
(0, ∞)
(−∞, ∞)
Answer: (a) x2 −4 > 0 and so x < −2 or x > 2,
thus the domain of f is (−∞, −2) ∪ (2, ∞).
If b > 1, f is increasing on its domain (0, ∞).
(b) −x > 0 and so x < 0. Thus the domain of
g is (−∞, 0).
If 0 < b < 1, f is decreasing on its domain
(0, ∞).
(c) 5 − x > 0 and x < 5. Thus the domain of f
is (−∞, 5).
5. (a) Describe how the graph of y = log2 (−x)
relates to the graph of y = log2 x.
tained by reflecting the graph of y = log2 x over
the y-axis.
(b) Describe how the graph of y = 5−log2 (x−3)
relates to the graph of y = log2 x.
(b) The graph of y = 5 − log2 (x − 3) is obtained
by shifting the graph of y = log2 x 3 units to
the right, then relfecting it over the x-axis, and
then shifting that graph 5 units up.
Answer: (a) The graph of y = log2 (−x) is ob-