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Prof. Fowler
MAT 106: Trigonometry
Quiz on Sections 5.1-5.3, Version A: Inverse, Exponential, and Logarithmic Functions – Solutions
A. 1. 3
3 2
8 8 82
3 2
8 8 8
8 8 8
5log log 5 log
log 5 log log
log 5 3 log 2 log
tt w
w
t w
t w
2.
6 3
6 3
6
3
6 log log 3 log 6 log log 3 log
log log log
log log
log
x y z x y z
x y z
x y z
x
y z
B. 3.
3 2
4
3 2
4
3 2
4
4 3 2
4 3 2
2 3 4
2 3 4
3 4
2
xf x
x
xy
x
yx
y
x y y
xy x y
xy y x
y x x
xy
x
1 3 4
2
xf x
x
4.
5
5
5
5
4
4
4
4
ln 4 5
ln 4
5
x
x
y
y
h x e
y e
x e
x e
x y
xy
1
ln 4
5
xh x
OR 1 1
ln 45
h x x
Prof. Fowler
5.
2
2
2
log 3
log 3
log 3
2 3
2 3
x
x
g x x
y x
x y
y
y
1 2 3xg x
C. 6. 4
ln 9 4
9
x
e x
4 9x e
7.
3
3
3
3
3
8log 3
27
8
27
27
8
27
8
27
8
x
x
x
x
x
3
2x
8. 2 3
2 3 5
10 100,000
10 10
2 3 5
2 8
x
x
x
x
4x
Prof. Fowler
Extra Credit
9. The domain of the inverse of 1f is simply the domain of f .
There are two possible approaches:
a. Logical reasoning based on logarithmic functions being inverses of exponential functions:
f involves a logarithmic function, which is the inverse of an exponential function.
The domain of a function is equivalent to the range of its inverse.
The range of any parent exponential function is 0,y .
Consequently, the domain of any parent logarithmic function is 0,x .
The domain of a function is affected by its horizontal transformations.
f involves only one horizontal transformation, which is a translation 7 units left.
Therefore, the domain of f is obtained by transforming 0,x accordingly:
70, 7,xx x
b. Work directly with domain restrictions on logarithmic functions:
7 0
7
x
x
7,x
10.
2 2
2
2
2
3 2
2
2
log 7 log 3 Domain: 7 0 AND 0 7
log 7 3
log 7 3
2 7
8 7
7 8 0
8 1 0
8 0 OR 1 0
8 OR 1
x x x x x
x x
x x
x x
x x
x x
x x
x x
x x
1 is not in the domain
8x
