Math 12 '- Precalculus '- Exponential And Logarithm Functions Test - 2009 Page 2
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1
2
3Math 12 – Precalculus – Exponential and Logarithm Functions Test Solutions – fall ’09
−
+
=
1
x
/ 2
1. Rewrite the equation
3.98 10
10
in equivalent logarithmic form, then solve for x.
Use a calculator to approximate x to the nearest ten thousandth.
(
)
(
)
(
)
−
−
+
=
⇔
=
⇔ −
=
⇔ =
−
1
x
/2
1
x
/2
3.98 10
10
10
6.02
1
x
/ 2 log 6.02
x
2 1 log 6.02
SOLN:
(
)
(
)
≈
−
=
≈
x
2 1 0.77960
2 0.22040
0.4408
(
)
−
−
=
2. Rewrite the equation
8 4 log
x
2
6.09
in equivalent exponential form, then solve for x.
Use a calculator to approximate x to the nearest ten thousandth.
SOLN:
(
)
(
)
(
)
−
−
=
⇔ −
−
= −
⇔
−
=
⇔ − =
≈
0.4775
8 4 log
x
2
6.09
4 log
x
2
1.91
log
x
2
0.4775
x
2 10
3.00262
≈
x
5.0026
3. Solve the equation for x. Approximate to 4 digits, if appropriate.
=
×
−
x
11
a.
10
6.67259 10
(
)
x =
−
≈
−
= −
SOLN:
log 6.67259
11 0.824294 11
10.175706
=
x
/24
b.
1.01
7
(
)
x
24 ln 7
(
)
= ⇔
=
⇔
=
⇔ =
≈
x
/24
x
/24
SOLN:
1.01
7
ln 1.01
ln 7
ln 1.01
ln 7
x
4693
(
)
24
ln 1.01
4. Use properties of logarithms to solve the equation. Approximate to 4 digits if appropriate.
( )
(
)
+
−
=
a.
log
x
log 44 3
x
7
2
2
(
)
( )
(
)
(
)
+
−
= ⇒
−
= ⇔ −
+
=
2
log
x
log 44 3
x
7
log
x
44 3
x
7
3
x
44
x
128
2
2
2
SOLN:
32
(
)(
)
⇔
−
−
= ⇔
−
−
= ⇔
=
=
2
3
x
44
x
128 0
3
x
32
x
4
0
x
or
x
4
3
⎛
⎞
10
( )
+
=
⎜
⎟
b.
log
log 1
0.5
SOLN:
⎝
⎠
2
9x
±
⎛
⎞
1/4
10
1
10
( )
+
=
⇔
−
+ =
⇔
=
⇔
=
⇔
=
2
2
2
⎜
⎟
log
log 1
0.5
log10 log 9
x
0 0.5
log 9
x
9
x
10
x
⎝
⎠
2
9
x
2
3
( )
f x = +
x
5. Given
1 1.322
.
( )
−
=
1
a. Find a formula for the inverse function,
y
f
x
.
(
)
(
)
(
)
= +
⇔
= − ⇔
=
−
⇔
=
−
x
x
x
SOLN:
y
1 1.322
1.322
y
1
log 1.322
log
y
1
x
log1.322 log
y
1
(
)
−
log
y
1
(
)
( )
(
)
=
≈
−
⇔
−
=
−
1
x
8.2487 log
y
1
f
x
8.2487 log
x
1
(
)
log 1.322
( )
b. Find the intercept and asymptote of
f x .
( )
=
SOLN: The graph of
y
f x
rises from a horizontal asymptote y = 1 through the
y-intercept at (0,2).
( )
−
1
c. Find the intercept and asymptote of
f
x
.
( )
=
−
1
SOLN: The graph of
y
f
x
rises along a vertical asymptote x = 1 through the
x-intercept at (2,0).
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