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Math Problems
Algebra 2
Quotient property of logarithms
Mason and Liam just raced each other in a
50
50
50
-yard dash, and it was really close! Mason ran the race in
7.75
7.75
7.75
seconds, and Liam ran it in
7
7
8
7\frac{7}{8}
7
8
7
seconds.
\newline
Who won the race?
\newline
Choices:
\newline
(A)Mason
\newline
(B)Liam
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Solve the following system of equations.
\newline
x
+
2
y
=
10
x+2y=10
x
+
2
y
=
10
\newline
−
x
−
5
y
=
−
22
-x-5y=-22
−
x
−
5
y
=
−
22
\newline
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Which of the following is equivalent to
log
(
a
b
)
\log\left(\frac{a}{b}\right)
lo
g
(
b
a
)
.
\newline
a)
log
a
log
b
\frac{\log a}{\log b}
l
o
g
b
l
o
g
a
\newline
b)
log
(
a
−
b
)
\log(a-b)
lo
g
(
a
−
b
)
\newline
c)
log
a
−
log
b
\log a-\log b
lo
g
a
−
lo
g
b
\newline
d)
log
b
−
log
a
\log b-\log a
lo
g
b
−
lo
g
a
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Use the quotient property of logs to write as a difference of logarithms. Simplify if possible.
\newline
log
(
100
y
)
\log\left(\frac{100}{y}\right)
lo
g
(
y
100
)
\newline
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M
⋅
V
=
P
⋅
Q
M \cdot V=P \cdot Q
M
⋅
V
=
P
⋅
Q
\newline
The quantitative theory of money states that given
M
M
M
dollars in circulation in a year, a monetary velocity of
V
V
V
, a price level of
P
P
P
, and a real output of
Q
Q
Q
dollars, the given equation is correct. An economist considers the case where the dollars in circulation and the real output are known constants. Which of the following expressions is the change in monetary velocity as the price level increases by
1
1
1
?
\newline
Choose
1
1
1
answer:
\newline
(A)
M
M
M
\newline
(B)
Q
Q
Q
\newline
(C)
M
Q
\frac{M}{Q}
Q
M
\newline
(D)
Q
M
\frac{Q}{M}
M
Q
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What is the inverse of the function
y
=
log
3
x
y=\log _{3} x
y
=
lo
g
3
x
?
\newline
(A)
y
=
x
3
y=x^{3}
y
=
x
3
\newline
(B)
y
=
log
x
3
y=\log _{x} 3
y
=
lo
g
x
3
\newline
(C)
y
=
3
x
y=3^{x}
y
=
3
x
\newline
(D)
x
=
3
y
x=3^{y}
x
=
3
y
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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
z
2
y
\log \frac{x z^{2}}{y}
lo
g
y
x
z
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
z
4
y
3
x
\log \frac{z^{4} \sqrt[3]{y}}{x}
lo
g
x
z
4
3
y
\newline
Answer:
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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
5
z
3
x
2
\log \frac{y^{5} \sqrt[3]{z}}{x^{2}}
lo
g
x
2
y
5
3
z
\newline
Answer:
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Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
5
y
2
z
3
\log \frac{x^{5}}{y^{2} z^{3}}
lo
g
y
2
z
3
x
5
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
5
z
y
4
\log \frac{x^{5} z}{y^{4}}
lo
g
y
4
x
5
z
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
z
4
y
4
3
\log \frac{x}{z^{4} \sqrt[3]{y^{4}}}
lo
g
z
4
3
y
4
x
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
2
y
3
z
\log \frac{x^{2} \sqrt[3]{y}}{z}
lo
g
z
x
2
3
y
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
2
z
2
3
x
5
\log \frac{y^{2} \sqrt[3]{z^{2}}}{x^{5}}
lo
g
x
5
y
2
3
z
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
\log x
lo
g
x
, and
log
y
\log y
lo
g
y
.
\newline
log
x
2
y
4
\log \frac{x^{2}}{y^{4}}
lo
g
y
4
x
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
\log x
lo
g
x
, and
log
y
\log y
lo
g
y
.
\newline
log
x
5
y
3
\log \frac{x^{5}}{y^{3}}
lo
g
y
3
x
5
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
\log x
lo
g
x
, and
log
y
\log y
lo
g
y
.
\newline
log
x
3
y
2
\log \frac{x^{3}}{y^{2}}
lo
g
y
2
x
3
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
\log x
lo
g
x
, and
log
y
\log y
lo
g
y
.
\newline
log
x
5
y
4
\log \frac{x^{5}}{y^{4}}
lo
g
y
4
x
5
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
\log x
lo
g
x
, and
log
y
\log y
lo
g
y
.
\newline
log
x
2
y
\log \frac{x^{2}}{y}
lo
g
y
x
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
5
x
z
4
3
\log \frac{y^{5} x}{\sqrt[3]{z^{4}}}
lo
g
3
z
4
y
5
x
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
z
5
x
\log \frac{y}{z^{5} x}
lo
g
z
5
x
y
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
z
5
y
\log \frac{x}{z^{5} y}
lo
g
z
5
y
x
\newline
Answer:
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
4
−
log
b
4
\log _{b} 4-\log _{b} 4
lo
g
b
4
−
lo
g
b
4
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
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Write the expression below as a single logarithm in simplest form.
\newline
log
b
9
−
log
b
9
\log _{b} 9-\log _{b} 9
lo
g
b
9
−
lo
g
b
9
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
10
+
log
b
8
\log _{b} 10+\log _{b} 8
lo
g
b
10
+
lo
g
b
8
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
4
log
b
2
+
log
b
5
4 \log _{b} 2+\log _{b} 5
4
lo
g
b
2
+
lo
g
b
5
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
3
+
log
b
2
\log _{b} 3+\log _{b} 2
lo
g
b
3
+
lo
g
b
2
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
3
2 \log _{b} 3
2
lo
g
b
3
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
9
2 \log _{b} 9
2
lo
g
b
9
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
5
log
b
2
−
log
b
4
5 \log _{b} 2-\log _{b} 4
5
lo
g
b
2
−
lo
g
b
4
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
8
2 \log _{b} 8
2
lo
g
b
8
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
5
2 \log _{b} 5
2
lo
g
b
5
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
4
log
b
2
4 \log _{b} 2
4
lo
g
b
2
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
5
+
log
b
4
\log _{b} 5+\log _{b} 4
lo
g
b
5
+
lo
g
b
4
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
3
+
log
b
6
2 \log _{b} 3+\log _{b} 6
2
lo
g
b
3
+
lo
g
b
6
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
7
+
log
b
9
\log _{b} 7+\log _{b} 9
lo
g
b
7
+
lo
g
b
9
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
3
log
b
4
3 \log _{b} 4
3
lo
g
b
4
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
3
log
b
3
3 \log _{b} 3
3
lo
g
b
3
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
6
+
log
b
7
\log _{b} 6+\log _{b} 7
lo
g
b
6
+
lo
g
b
7
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
3
−
log
b
9
2 \log _{b} 3-\log _{b} 9
2
lo
g
b
3
−
lo
g
b
9
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
6
−
log
b
2
\log _{b} 6-\log _{b} 2
lo
g
b
6
−
lo
g
b
2
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
5
log
b
2
5 \log _{b} 2
5
lo
g
b
2
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
10
−
log
b
10
\log _{b} 10-\log _{b} 10
lo
g
b
10
−
lo
g
b
10
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
log
b
2
+
2
log
b
2
\log _{b} 2+2 \log _{b} 2
lo
g
b
2
+
2
lo
g
b
2
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
4
2 \log _{b} 4
2
lo
g
b
4
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Condense the logarithm
\newline
5
log
b
+
3
log
r
5 \log b+3 \log r
5
lo
g
b
+
3
lo
g
r
\newline
Answer:
log
(
□
)
\log (\square)
lo
g
(
□
)
Get tutor help
Condense the logarithm
\newline
k
log
a
+
3
log
c
k \log a+3 \log c
k
lo
g
a
+
3
lo
g
c
\newline
Answer:
log
(
□
)
\log (\square)
lo
g
(
□
)
Get tutor help
Condense the logarithm
\newline
g
log
c
+
5
log
q
g \log c+5 \log q
g
lo
g
c
+
5
lo
g
q
\newline
Answer:
log
(
□
)
\log (\square)
lo
g
(
□
)
Get tutor help
Condense the logarithm
\newline
log
a
−
6
log
c
\log a-6 \log c
lo
g
a
−
6
lo
g
c
\newline
Answer:
log
(
□
)
\log (\square)
lo
g
(
□
)
Get tutor help
Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
3
=
1
3
\log _{x} 3=\frac{1}{3}
lo
g
x
3
=
3
1
\newline
Answer:
Get tutor help
1
2
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