Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 4
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41Worksheet # 3: The Exponential Function and the Logarithm
1. (a) Graph the functions f (x) = 2 and g(x) = 2
and give the domains and range of each function.
(b) Determine if each function is one-to-one. Determine if each function is increasing or decreasing.
(c) Graph the inverse function to f . Give the domain and range of the inverse function.
2 +1
3
2. (a) Solve 10
= 100.
(f) Solve e
= 3.
(
)
(b) Solve 2
= 16.
(g) Solve 4 = e.
+2
(c) Solve 2 = 4
.
(d) Find log
(8).
(h) Solve ln(x + 1) + ln(x
1) = ln(3). Be sure
2
2
(e) Find ln(e
).
to check your answer.
(3 )
3
2
3. Evaluate the expressions 4
and (4
)
. Are they equal?
2
4. Suppose a and b are positive real numbers and ln(ab) = 3 and ln(ab
) = 5. Find ln(a), ln(b), and
3
ln(a
/ b).
5. Consider the function f (x) = 1 + ln(x). Determine the inverse function of f . Give the domain and
1
range of f and of the inverse function f
.
6. Suppose that a population doubles every two hours. If we have one hundred critters at 12 noon, how
many will there be after 1 hour? after 2 hours? How many were there at 11am? Give a formula for
the number of critters at t hours after 12 noon.
7. Let f be the function f (x) = 4 . Write the function f in the form f (x) = e .
8. Let f be the function f (x) = 5 3 . Write the function in the form Ae .
9. Suppose that f is a function of the form f (x) = Ae . If f (2) = 20 and f (5) = 10, will we have k > 0
or k < 0? Find A and k so that f (2) = 20 and f (5) = 10.
1
10. Let f be the function given by f (x) = tan(x) with x in the interval (π/2, 3π/2). Find f
(0) and
1
f
( 3).
1
1
1
Sketch the graphs of f and f
. Give a formula which expresses f
in terms of the function tan
(or arctan).
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