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Byju's Answer
Standard XII
Mathematics
Solving Homogeneous Differential Equations
The value of ...
Question
The value of
log
x
+
log
(
1
+
1
x
)
+
log
(
1
+
1
1
+
x
)
+
log
(
1
+
1
2
+
x
)
+
…
⋯
+
log
(
1
+
1
(
n
−
1
+
x
)
)
A
log
x
n
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B
log
n
x
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C
log
(
n
+
x
)
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D
log
(
n
−
1
)
x
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Solution
The correct option is
C
log
(
n
+
x
)
log
x
+
log
(
1
+
1
x
)
+
log
(
1
+
1
1
+
x
)
+
log
(
1
+
1
2
+
x
)
+
…
…
…
+
log
(
1
+
1
(
n
−
1
+
x
)
)
Multiplying first
2
terms and using property
log
n
+
log
m
=
log
(
m
n
)
:
=
log
(
x
+
1
)
+
log
(
1
+
1
1
+
x
)
+
log
(
1
+
1
2
+
x
)
+
…
…
…
+
log
(
1
+
1
(
n
−
1
+
x
)
)
Multiplying first
2
terms again:
=
log
(
x
+
2
)
+
log
(
1
+
1
2
+
x
)
+
…
…
…
+
log
(
1
+
1
(
n
−
1
+
x
)
)
Continuing this series we get:
=
log
(
n
−
1
+
x
)
+
log
(
1
+
1
(
n
−
1
+
x
)
)
=
log
(
n
+
x
)
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Similar questions
Q.
The value of
log
x
+
log
(
1
+
1
x
)
+
log
(
1
+
1
1
+
x
)
+
log
(
1
+
1
2
+
x
)
+
…
…
…
+
log
(
1
+
1
(
n
−
1
+
x
)
)
Q.
The value of
∫
1
x
+
x
log
x
d
x
is
(a) 1 + log x
(b) x + log x
(c) x log (1 + log x)
(d) log (1 + log x)
Q.
Find the value of x satisfying the equation
log
1
2
(
x
−
1
)
+
log
1
2
(
x
+
1
)
−
log
1
√
2
(
7
−
x
)
=
1
Q.
The sum of values of
x
satisfying
log
1
/
2
(
x
−
1
)
+
log
1
/
2
(
x
+
1
)
−
log
1
/
√
2
(
7
−
x
)
=
1
is
Q.
l
o
g
(
1
+
x
)
−
l
o
g
(
1
−
x
)
=
1
Find the value of x.
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