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Algebra 2
Convert between exponential and logarithmic form: all bases
Write the exponential equation in logarithmic form.
e
2
5
≈
1.2214
e^{\frac{2}{5}} \approx 1.2214
e
5
2
≈
1.2214
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Write the exponential equation in logarithmic form.
\newline
e
3.2
≈
24.5325
e^{3.2} \approx 24.5325
e
3.2
≈
24.5325
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Write the exponential equation in logarithmic form.
\newline
e
2.5
≈
12.1825
e^{2.5} \approx 12.1825
e
2.5
≈
12.1825
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Write the exponential equation in logarithmic form.
\newline
e
3
4
≈
2.117
e^{\frac{3}{4}} \approx 2.117
e
4
3
≈
2.117
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Write the exponential equation in logarithmic form.
\newline
e
6
≈
403.429
e^6 \approx 403.429
e
6
≈
403.429
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Write the exponential equation in logarithmic form.
\newline
e
1
≈
2.718
e^1 \approx 2.718
e
1
≈
2.718
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Write the exponential equation in logarithmic form.
\newline
e
2
≈
7.389
e^2 \approx 7.389
e
2
≈
7.389
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Write the exponential equation in logarithmic form.
\newline
e
3
≈
20.085
e^3 \approx 20.085
e
3
≈
20.085
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Write the exponential equation in logarithmic form.
\newline
e
5
≈
148.413
e^5 \approx 148.413
e
5
≈
148.413
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Write the exponential equation in logarithmic form.
1
6
3
2
=
64
\newline 16^{\frac{3}{2}} = 64
1
6
2
3
=
64
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Convert the exponential equation in logarithmic form.
\newline
8
4
3
=
16
8^{\frac{4}{3}} = 16
8
3
4
=
16
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Convert the exponential equation in logarithmic form.
\newline
9
0
=
1
9^0 = 1
9
0
=
1
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Convert the exponential equation in logarithmic form.
\newline
1
5
−
1
=
1
15
15^{-1} = \frac{1}{15}
1
5
−
1
=
15
1
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Convert the exponential equation in logarithmic form.
\newline
1
0
−
2
=
1
100
10^{-2} = \frac{1}{100}
1
0
−
2
=
100
1
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Convert the exponential equation in logarithmic form.
\newline
1
3
2
=
169
13^2 = 169
1
3
2
=
169
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Convert the exponential equation in logarithmic form.
\newline
1
0
2
=
100
10^2 = 100
1
0
2
=
100
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Convert
ln
(
7
)
=
a
\ln(7) = a
ln
(
7
)
=
a
to its exponential form.
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Write the logarithmic equation in exponential form.
\newline
log
7
373
=
3
\log_7 373 = 3
lo
g
7
373
=
3
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What is the domain of this exponential function
?
?
?
\newline
y
=
7
(
x
+
4
)
y = 7^{(x+4)}
y
=
7
(
x
+
4
)
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What is the domain of this exponential function
?
?
?
\newline
y
=
5
x
+
5
y = 5^{x+5}
y
=
5
x
+
5
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Consider the equation
−
5
⋅
e
10
t
=
−
30
-5 \cdot e^{10 t}=-30
−
5
⋅
e
10
t
=
−
30
.
\newline
Solve the equation for
t
t
t
. Express the solution as a logarithm in base-
e
e
e
.
\newline
t
=
□
t = \square
t
=
□
\newline
Approximate the value of
t
t
t
. Round your answer to the nearest thousandth.
\newline
t
≈
□
t \approx \square
t
≈
□
\newline
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Consider the equation
14
⋅
1
0
0.5
w
=
100
14 \cdot 10^{0.5 w}=100
14
⋅
1
0
0.5
w
=
100
.
\newline
Solve the equation for
w
w
w
. Express the solution as a logarithm in base-
10
10
10
.
\newline
w
=
□
w = \square
w
=
□
\newline
Approximate the value of
w
w
w
. Round your answer to the nearest thousandth.
\newline
w
≈
□
w \approx \square
w
≈
□
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5
6
W
=
10
3
\frac{5}{6} W = \frac{10}{3}
6
5
W
=
3
10
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
ln
(
x
2
−
2
x
+
7
)
=
9
5
\ln \left(x^{2}-2 x+7\right)=\frac{9}{5}
ln
(
x
2
−
2
x
+
7
)
=
5
9
\newline
Answer:
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
log
(
x
2
+
3
x
−
6
)
=
2
\log \left(x^{2}+3 x-6\right)=2
lo
g
(
x
2
+
3
x
−
6
)
=
2
\newline
Answer:
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
ln
(
x
2
+
3
x
−
18
)
=
2
\ln \left(x^{2}+3 x-18\right)=2
ln
(
x
2
+
3
x
−
18
)
=
2
\newline
Answer:
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
ln
(
x
2
+
4
x
+
11
)
=
2
\ln \left(x^{2}+4 x+11\right)=2
ln
(
x
2
+
4
x
+
11
)
=
2
\newline
Answer:
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
ln
(
2
x
+
5
)
=
x
\ln (2 x+5)=x
ln
(
2
x
+
5
)
=
x
\newline
Answer:
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Write the log equation as an exponential equation. You do not need to solve for
x
\mathrm{x}
x
.
\newline
log
5
(
x
2
−
2
x
+
20
)
=
5
x
\log _{5}\left(x^{2}-2 x+20\right)=5 x
lo
g
5
(
x
2
−
2
x
+
20
)
=
5
x
\newline
Answer:
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Write
6
5
=
7776
6^{5}=7776
6
5
=
7776
in logarithmic form
\newline
(A)
log
6
5
=
7776
\log_{6}5=7776
lo
g
6
5
=
7776
\newline
(B)
log
6
7776
=
5
\log_{6}7776=5
lo
g
6
7776
=
5
\newline
(C)
log
5
6
=
7776
\log_{5}6=7776
lo
g
5
6
=
7776
\newline
(D)
log
5
7776
=
6
\log_{5}7776=6
lo
g
5
7776
=
6
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Write an exponential function in the form
y
=
a
b
x
y=a b^{x}
y
=
a
b
x
that goes through the points
(
0
,
14
)
(0,14)
(
0
,
14
)
and
(
7
,
1792
)
(7,1792)
(
7
,
1792
)
.
\newline
Answer:
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log
81
3
=
\log _{81} 3=
lo
g
81
3
=
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log
4
4
=
\log _{4} 4=
lo
g
4
4
=
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log
10
1
,
000
=
\log _{10} 1,000=
lo
g
10
1
,
000
=
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log
2
256
=
\log _{2} 256=
lo
g
2
256
=
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log
10
10
,
000
=
\log _{10} 10,000=
lo
g
10
10
,
000
=
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Rewrite the following in the form
log
(
c
)
\log (c)
lo
g
(
c
)
.
\newline
log
(
2
)
+
log
(
2
)
\log (2)+\log (2)
lo
g
(
2
)
+
lo
g
(
2
)
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log
10
1
,
000
,
000
=
\log _{10} 1,000,000=
lo
g
10
1
,
000
,
000
=
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log
3
243
=
\log _{3} 243=
lo
g
3
243
=
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log
4
64
=
\log _{4} 64=
lo
g
4
64
=
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log
2
4
=
\log _{2} 4=
lo
g
2
4
=
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log
5
625
=
\log _{5} 625=
lo
g
5
625
=
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log
8
512
=
\log _{8} 512=
lo
g
8
512
=
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log
2
128
=
\log _{2} 128=
lo
g
2
128
=
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Of all eligible voters in your county,
70
%
70 \%
70%
are currently registered to vote.
\newline
A recent poll predicts that the relationship between
V
V
V
, the percentage of all eligible voters who are registered to vote, and
t
t
t
, the number of years from now will be modeled by the following equation.
\newline
V
=
100
−
30
⋅
e
−
0.04
t
V=100-30 \cdot e^{-0.04 t}
V
=
100
−
30
⋅
e
−
0.04
t
\newline
In how many years will
80
%
80 \%
80%
of all eligible voters in your county be registered to vote?
\newline
Give an exact answer expressed as a natural logarithm.
\newline
years
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Noah borrows
$
2000
\$ 2000
$2000
from his father and agrees to repay the loan and any interest determined by his father as soon as he has the money.
\newline
The relationship between the amount of money,
A
A
A
, in dollars that Noah owes his father (including interest), and the elapsed time,
t
t
t
, in years, is modeled by the following equation.
\newline
A
=
2000
e
0.1
t
A=2000 e^{0.1 t}
A
=
2000
e
0.1
t
\newline
How long did it take Noah to pay off his loan if the amount he paid to his father was equal to
$
2450
\$ 2450
$2450
? Give an exact answer expressed as a natural logarithm.
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A scientist measures the initial amount of Carbon
−
14
-14
−
14
in a substance to be
25
25
25
grams.
\newline
The relationship between
A
A
A
, the amount of Carbon
−
14
-14
−
14
remaining in that substance, in grams, and
t
t
t
, the elapsed time, in years, since the initial measurement is modeled by the following equation.
\newline
A
=
25
e
−
0.00012
t
A=25 e^{-0.00012 t}
A
=
25
e
−
0.00012
t
\newline
In how many years will the substance contain exactly
20
20
20
grams
(
g
)
(\mathrm{g})
(
g
)
of Carbon
−
14
-14
−
14
?
\newline
Give an exact answer expressed as a natural logarithm.
\newline
years
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Write the expression in exponential form.
\newline
9
×
9
×
9
×
9
×
9
×
9
×
9
=
9 \times 9 \times 9 \times 9 \times 9 \times 9 \times 9=
9
×
9
×
9
×
9
×
9
×
9
×
9
=
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Write the expression in exponential form.
\newline
7
×
7
×
7
×
7
×
7
×
7
=
7 \times 7 \times 7 \times 7 \times 7 \times 7=
7
×
7
×
7
×
7
×
7
×
7
=
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Write the expression in exponential form.
\newline
6
×
6
×
6
×
6
×
6
=
6 \times 6 \times 6 \times 6 \times 6=
6
×
6
×
6
×
6
×
6
=
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