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Math Problems
Precalculus
Quotient property of logarithms
Find the numerical value of the log expression.
\newline
log
a
=
9
log
b
=
6
log
c
=
4
log
b
5
c
5
a
9
\begin{array}{c} \log a=9 \quad \log b=6 \quad \log c=4 \\ \log \frac{\sqrt{b^{5} c^{5}}}{a^{9}} \end{array}
lo
g
a
=
9
lo
g
b
=
6
lo
g
c
=
4
lo
g
a
9
b
5
c
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
4
log
b
=
−
8
log
c
=
−
3
log
a
3
b
5
c
5
\begin{array}{c} \log a=4 \quad \log b=-8 \quad \log c=-3 \\ \log \frac{\sqrt{a^{3} b^{5}}}{c^{5}} \end{array}
lo
g
a
=
4
lo
g
b
=
−
8
lo
g
c
=
−
3
lo
g
c
5
a
3
b
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
7
log
b
=
4
log
c
=
6
log
b
8
a
6
c
5
\begin{array}{c} \log a=7 \quad \log b=4 \quad \log c=6 \\ \log \frac{b^{8}}{a^{6} c^{5}} \end{array}
lo
g
a
=
7
lo
g
b
=
4
lo
g
c
=
6
lo
g
a
6
c
5
b
8
\newline
Answer:
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Write the expression below as a single logarithm in simplest form.
\newline
3
log
b
3
−
log
b
3
3 \log _{b} 3-\log _{b} 3
3
lo
g
b
3
−
lo
g
b
3
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
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Simplify the fraction completely. If the fraction does not simplify, submit the fraction in its current form.
\newline
11
x
4
66
\frac{11 x^{4}}{66}
66
11
x
4
\newline
Answer:
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log
2
(
5
)
+
log
2
(
2
)
=
?
?
\log _{2}(5)+\log _{2}(2)=? ?
lo
g
2
(
5
)
+
lo
g
2
(
2
)
=
??
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
729
=
6
5
\log _{x} 729=\frac{6}{5}
lo
g
x
729
=
5
6
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
5
(
7
x
)
+
2
log
5
(
5
)
=
4
\log _{5}(7 x)+2 \log _{5}(5)=4
lo
g
5
(
7
x
)
+
2
lo
g
5
(
5
)
=
4
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
8
x
)
−
log
4
(
8
)
=
3
\log _{4}(8 x)-\log _{4}(8)=3
lo
g
4
(
8
x
)
−
lo
g
4
(
8
)
=
3
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
7
(
5
x
)
−
3
log
7
(
2
)
=
0
\log _{7}(5 x)-3 \log _{7}(2)=0
lo
g
7
(
5
x
)
−
3
lo
g
7
(
2
)
=
0
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
3
(
6
x
)
+
log
3
(
2
)
=
3
\log _{3}(6 x)+\log _{3}(2)=3
lo
g
3
(
6
x
)
+
lo
g
3
(
2
)
=
3
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
9
x
)
−
4
log
4
(
3
)
=
0
\log _{4}(9 x)-4 \log _{4}(3)=0
lo
g
4
(
9
x
)
−
4
lo
g
4
(
3
)
=
0
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
3
x
)
−
2
log
4
(
2
)
=
2
\log _{4}(3 x)-2 \log _{4}(2)=2
lo
g
4
(
3
x
)
−
2
lo
g
4
(
2
)
=
2
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
2
(
9
x
)
−
log
2
(
2
)
=
5
\log _{2}(9 x)-\log _{2}(2)=5
lo
g
2
(
9
x
)
−
lo
g
2
(
2
)
=
5
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
9
x
)
+
log
4
(
9
)
=
3
\log _{4}(9 x)+\log _{4}(9)=3
lo
g
4
(
9
x
)
+
lo
g
4
(
9
)
=
3
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
2
x
)
−
3
log
4
(
9
)
=
0
\log _{4}(2 x)-3 \log _{4}(9)=0
lo
g
4
(
2
x
)
−
3
lo
g
4
(
9
)
=
0
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
7
x
)
+
2
log
4
(
8
)
=
3
\log _{4}(7 x)+2 \log _{4}(8)=3
lo
g
4
(
7
x
)
+
2
lo
g
4
(
8
)
=
3
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
4
(
5
x
)
+
log
4
(
7
)
=
3
\log _{4}(5 x)+\log _{4}(7)=3
lo
g
4
(
5
x
)
+
lo
g
4
(
7
)
=
3
\newline
Answer:
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Solve for the exact value of
x
x
x
.
\newline
log
6
(
4
x
)
+
log
6
(
9
)
=
3
\log _{6}(4 x)+\log _{6}(9)=3
lo
g
6
(
4
x
)
+
lo
g
6
(
9
)
=
3
\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
4
x
(
4
x
)
=
9
8
\log _{4 x}(4 x)=\frac{9}{8}
lo
g
4
x
(
4
x
)
=
8
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
3
x
(
4
x
)
=
4
9
\log _{3 x}(4 x)=\frac{4}{9}
lo
g
3
x
(
4
x
)
=
9
4
\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
2
x
(
2
)
=
5
3
\log _{2 x}(2)=\frac{5}{3}
lo
g
2
x
(
2
)
=
3
5
\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
+
5
)
(
2
x
)
=
6
7
\log _{(x+5)}(2 x)=\frac{6}{7}
lo
g
(
x
+
5
)
(
2
x
)
=
7
6
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
9
x
=
−
1
2
\log _{9} x=-\frac{1}{2}
lo
g
9
x
=
−
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
16
x
=
−
5
4
\log _{16} x=-\frac{5}{4}
lo
g
16
x
=
−
4
5
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
100
x
=
−
1
2
\log _{100} x=-\frac{1}{2}
lo
g
100
x
=
−
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
216
=
3
2
\log _{x} 216=\frac{3}{2}
lo
g
x
216
=
2
3
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
64
x
=
−
1
3
\log _{64} x=-\frac{1}{3}
lo
g
64
x
=
−
3
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
25
=
2
3
\log _{x} 25=\frac{2}{3}
lo
g
x
25
=
3
2
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
5
=
1
2
\log _{x} 5=\frac{1}{2}
lo
g
x
5
=
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
16
x
=
−
1
2
\log _{16} x=-\frac{1}{2}
lo
g
16
x
=
−
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
343
=
3
2
\log _{x} 343=\frac{3}{2}
lo
g
x
343
=
2
3
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
64
=
6
5
\log _{x} 64=\frac{6}{5}
lo
g
x
64
=
5
6
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
1
8
=
−
3
4
\log _{x} \frac{1}{8}=-\frac{3}{4}
lo
g
x
8
1
=
−
4
3
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
27
x
=
−
2
3
\log _{27} x=-\frac{2}{3}
lo
g
27
x
=
−
3
2
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
25
x
=
1
2
\log _{25} x=\frac{1}{2}
lo
g
25
x
=
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
16
x
=
1
2
\log _{16} x=\frac{1}{2}
lo
g
16
x
=
2
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
10
=
1
3
\log _{x} 10=\frac{1}{3}
lo
g
x
10
=
3
1
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
125
x
=
−
4
3
\log _{125} x=-\frac{4}{3}
lo
g
125
x
=
−
3
4
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
x
1
4
=
−
2
5
\log _{x} \frac{1}{4}=-\frac{2}{5}
lo
g
x
4
1
=
−
5
2
\newline
Answer:
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Solve the following logarithm problem for the positive solution for
x
x
x
.
\newline
log
125
x
=
2
3
\log _{125} x=\frac{2}{3}
lo
g
125
x
=
3
2
\newline
Answer:
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For any positive numbers
a
,
b
a, b
a
,
b
and
d
d
d
, with
b
≠
1
b \neq 1
b
=
1
,
\newline
log
b
(
a
d
)
=
log
b
a
−
log
b
d
\log _{b}\left(\frac{a}{d}\right)=\log _{b} a-\log _{b} d
lo
g
b
(
d
a
)
=
lo
g
b
a
−
lo
g
b
d
\newline
Examples:
\newline
log
(
2
5
)
=
log
2
−
log
5
ln
6
−
ln
2
=
ln
(
6
2
)
=
ln
3
\begin{array}{l} \log \left(\frac{2}{5}\right)=\log 2-\log 5 \\ \ln 6-\ln 2=\ln \left(\frac{6}{2}\right)=\ln 3 \end{array}
lo
g
(
5
2
)
=
lo
g
2
−
lo
g
5
ln
6
−
ln
2
=
ln
(
2
6
)
=
ln
3
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Evaluate.
\newline
log
7
1
4
\log _{7} \frac{1}{4}
lo
g
7
4
1
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If
f
(
x
)
=
log
2
(
4
3
x
6
+
x
4
+
8
)
f(x) = \log_2(4\sqrt{3}x^6 + x^4 + 8)
f
(
x
)
=
lo
g
2
(
4
3
x
6
+
x
4
+
8
)
, find
f
′
(
x
)
f'(x)
f
′
(
x
)
. Use exact values.
f
′
(
x
)
f'(x)
f
′
(
x
)
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Solve the following for
x
x
x
.
\newline
log
3
(
x
2
−
24
)
=
log
3
(
5
x
)
\log_{3}(x^{2}-24)=\log_{3}(5x)
lo
g
3
(
x
2
−
24
)
=
lo
g
3
(
5
x
)
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log
9
5
+
log
9
8
=
\log_{9}5+\log_{9}8=
lo
g
9
5
+
lo
g
9
8
=
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Evaluate:
\newline
log
25
(
1
125
)
\log_{25}\left(\frac{1}{125}\right)
lo
g
25
(
125
1
)
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Write as the sum and/or difference of logarithms. Express powers as factors.
\newline
log
16
(
19
m
n
)
\log_{16}\left(\frac{19\sqrt{m}}{n}\right)
lo
g
16
(
n
19
m
)
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Select the answer which is equivalent to the given expression using your calculator.
\newline
log
49
343
\log _{49} 343
lo
g
49
343
\newline
3
2
\frac{3}{2}
2
3
\newline
2
5
\frac{2}{5}
5
2
\newline
2
3
\frac{2}{3}
3
2
\newline
5
2
\frac{5}{2}
2
5
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What is the value of
log
5
1
5
\log _{5} \frac{1}{5}
lo
g
5
5
1
?
\newline
Answer:
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