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Math Problems
Algebra 2
Properties of logarithms: mixed review
Consider the equation
−
5
⋅
e
10
t
=
−
30
-5\cdot e^{10t}=-30
−
5
⋅
e
10
t
=
−
30
.Solve the equation for
t
t
t
. Express the solution as a logarithm in base-
e
e
e
.
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Consider the equation
4
⋅
1
0
−
3
x
=
18
4\cdot 10^{-3x}=18
4
⋅
1
0
−
3
x
=
18
. Solve the equation for
x
x
x
. Express the solution as a logarithm in base-
10
10
10
.
x
=
x=
x
=
Approximate the value of
x
x
x
. Round your answer to the nearest thousandth.
x
≈
x\approx
x
≈
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He predicts that the relationship between
N
N
N
, the number of branches on the tree, and
t
t
t
years, since the tree was planted can be modeled by the following equation.
N
=
5
⋅
100.
3
t
N = 5 \cdot 100.3^t
N
=
5
⋅
100.
3
t
According to Takumi's model, in how many years will the tree have
100
100
100
branches? Give an exact answer expressed as a base-
10
10
10
logarithm.
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Write a power represented with a positive base and a positive exponent whose value is less than the base.
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
3
log
3
(
7
z
)
)
\left(3^{\log _{3}(7 \sqrt{z})}\right)
(
3
l
o
g
3
(
7
z
)
)
\newline
Answer:
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Consider the equation
\newline
−
16
×
1
0
6
x
=
−
80
-16 \times 10^{6x} = -80
−
16
×
1
0
6
x
=
−
80
.
\newline
Solve the equation for
\newline
x
x
x
. Express the solution as a logarithm in base
−
10
-10
−
10
.
\newline
x
=
x=
x
=
\newline
Approximate the value of
\newline
x
x
x
. Round your answer to the nearest thousandth.
\newline
x
≈
x \approx
x
≈
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Let
h
(
x
)
=
log
(
x
)
h(x)=\log (x)
h
(
x
)
=
lo
g
(
x
)
.
\newline
Note: Here, we are referring to log base
10
10
10
.
\newline
Find
h
′
′
(
x
)
h^{\prime \prime}(x)
h
′′
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
1
x
2
-\frac{1}{x^{2}}
−
x
2
1
\newline
(B)
−
log
(
x
)
ln
(
10
)
-\frac{\log (x)}{\ln (10)}
−
l
n
(
10
)
l
o
g
(
x
)
\newline
(C)
−
1
x
2
ln
(
10
)
-\frac{1}{x^{2} \ln (10)}
−
x
2
l
n
(
10
)
1
\newline
(D)
log
(
x
)
\log (x)
lo
g
(
x
)
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Let
h
(
x
)
=
log
(
x
)
h(x)=\log (x)
h
(
x
)
=
lo
g
(
x
)
.
\newline
Note: Here, we are referring to log base
10
10
10
.
\newline
Find
h
′
′
(
x
)
h^{\prime \prime}(x)
h
′′
(
x
)
.
\newline
Choose
1
1
1
answer:
\newline
(A)
−
log
(
x
)
ln
(
10
)
-\frac{\log (x)}{\ln (10)}
−
l
n
(
10
)
l
o
g
(
x
)
\newline
(B)
−
1
x
2
ln
(
10
)
-\frac{1}{x^{2} \ln (10)}
−
x
2
l
n
(
10
)
1
\newline
(C)
−
1
x
2
-\frac{1}{x^{2}}
−
x
2
1
\newline
(D)
log
(
x
)
\log (x)
lo
g
(
x
)
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Expand the logarithm. Assume all expressions exist and are well-defined.
\newline
Write your answer as a sum or difference of base-
6
6
6
logarithms or multiples of base-
6
6
6
logarithms. The inside of each logarithm must be a distinct constant or variable.
\newline
log
6
w
6
\log_6 w^6
lo
g
6
w
6
\newline
______
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