Mathematics - Logarithms and Their Properties
Exam Duration: 45 Mins Total Questions : 30
If log23=a, log35=b and log7c=c, then the logarithm of the number 63 to base 140 is
- (a)
\(\frac { 1+2ac }{ 2c+abc+1 } \)
- (b)
\(\frac { 1-2ac }{ 2c-abc-1 } \)
- (c)
\(\frac { 1-2ac }{ 2c+abc+1 } \)
- (d)
\(\frac { 1+2ac }{ 2c-abc-1 } \)
The number of real values of the parameter \(\lambda\)for which (log16x)2-log16x+log16\(\lambda\)=0 with real coefficients will have exactly one solution is
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If \(y={ 2 }^{ \frac { 1 }{ \log _{ x }{ 4 } } }\) , then x is equal to
- (a)
\(\sqrt { y } \)
- (b)
y
- (c)
y2
- (d)
y3
if log2x+log2y\(\ge\)6, then the least value of x+y is
- (a)
4
- (b)
8
- (c)
16
- (d)
32
A rational number which is 50 times its own logarithm to the base 10 is
- (a)
1
- (b)
10
- (c)
100
- (d)
1000
If x=log5(1000) and y=log7(2058), then
- (a)
x>y
- (b)
x<y
- (c)
x=y
- (d)
none of these
If log102,log10(2x+1), log10(2x+3) are in AP, then
- (a)
x=0
- (b)
x=1
- (c)
x=log102
- (d)
\(x=\frac { 1 }{ 2 } \log _{ 2 }{ 5 } \)
\(7\log { \left( \frac { 16 }{ 15 } \right) } +5\log { \left( \frac { 25 }{ 24 } \right) } +3\log { \left( \frac { 81 }{ 80 } \right) } \) is equal to
- (a)
0
- (b)
1
- (c)
log 2
- (d)
log 3
If x=log35, y=log1725, which one of the following is correct?
- (a)
x
- (b)
x=y
- (c)
x>y
- (d)
None of these
The number log27 is
- (a)
an integer
- (b)
a rational number
- (c)
an irrational number
- (d)
a prime number
The value of \(\frac { 1 }{ \log _{ 2 }{ n } } +\frac { 1 }{ \log _{ 3 }{ n } } +....+\frac { 1 }{ \log _{ 43 }{ n } } \)is
- (a)
\(\frac { 1 }{ \log _{ 43! }{ n } } \)
- (b)
\(\frac { 1 }{ \log _{ 43 }{ n } } \)
- (c)
\(\frac { 1 }{ \log _{ 42 }{ n } } \)
- (d)
\(\frac { 1 }{ \log _{ 43 }{ n! } } \)
log7log7\(\sqrt { 7\sqrt { (7\sqrt { 7 } ) } } \) is equal to
- (a)
3log27
- (b)
3log22
- (c)
1-3log22
- (d)
1-3log27
If\(\frac { \log { x } }{ b-c } =\frac { \log { y } }{ c-a } =\frac { \log { z } }{ a-b } \) then xaybzc is equal to
- (a)
xyz
- (b)
abc
- (c)
0
- (d)
1
If \(\frac { x(y+z-x) }{ \log { x } } +\frac { y(z+x-y) }{ \log { y } } +\frac { z(x+y-z) }{ \log { z } } \)then xyyx=zyyz is equal to
- (a)
zxxz
- (b)
xzyx
- (c)
xyyz
- (d)
xxyy
If log32,log3(2x-5),log3(2x-7/2) are in AP, then x is equal to
- (a)
1
- (b)
2
- (c)
3
- (d)
4
If \(y={ a }^{ \frac { 1 }{ 1-\log _{ a }{ x } } }\quad and\quad z=y={ a }^{ \frac { 1 }{ 1-\log _{ a }{ y } } },\)then x is equal to
- (a)
\({ a }^{ \frac { 1 }{ 1+\log _{ a }{ z } } }\quad \)
- (b)
\({ a }^{ \frac { 1 }{ 1+\log _{ a }{ z } } }\quad \)
- (c)
\({ a }^{ \frac { 1 }{ 1-\log _{ a }{ z } } }\quad \)
- (d)
\({ a }^{ \frac { 1 }{ 2-\log _{ a }{ z } } }\quad \)
If \(In\left( \frac { a+b }{ 3 } \right) =\left( \frac { Ina+Inb }{ 2 } \right) ,\)then \(\frac { a }{ b } +\frac { b }{ a } \) is equal to
- (a)
1
- (b)
3
- (c)
5
- (d)
7
If log3{5+4log3(x-1)}=2, then x is equal to
- (a)
2
- (b)
4
- (c)
8
- (d)
log216
If 2xlog43+3log4x=27, then x is equal to
- (a)
2
- (b)
4
- (c)
8
- (d)
16
The interval of x in which the inequality \({ 5 }^{ \frac { 1 }{ 4 } \left( { log }_{ 5 }^{ 2 }\quad x \right) }\ge { 5x }^{ \frac { 1 }{ 5 } \left( { log }_{ 5 }x \right) }\)
- (a)
\((0,{ 5 }^{ -2\sqrt { 5 } }]\)
- (b)
\([{ 5 }^{ 2\sqrt { 5 } },\infty )\)
- (c)
both (a) and (b)
- (d)
none of these
The least value of expression 2log10x-logx0.01 is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
The solution of the equation \(\log _{ 7 }{ \log _{ 5 }{ \left( \sqrt { x+5 } +\sqrt { x } \right) } } =0\)is
- (a)
1
- (b)
3
- (c)
4
- (d)
5
The number of solutions of log4(x-1)=log2(x-3) is
- (a)
3
- (b)
1
- (c)
2
- (d)
0
If \(\frac { \log _{ 2 }{ x } }{ 4 } =\frac { \log _{ 2 }{ y } }{ 6 } =\frac { \log _{ 2 }{ z } }{ 3k } \)and x3y2z=1, then k is equal
- (a)
-8
- (b)
-4
- (c)
0
- (d)
\(\log _{ 2 }{ \left( \frac { 1 }{ 256 } \right) } \)
The expression \({ 5 }^{ \log _{ 1/5 }{ \left( 1/2 \right) } }+\log _{ \sqrt { 2 } }{ \left( \frac { 4 }{ \sqrt { 7 } +\sqrt { 3 } } \right) } +\log _{ 1/2 }{ \left( \frac { 1 }{ 10+2\sqrt { 21 } } \right) } \) simplifies to
- (a)
6
- (b)
4
- (c)
\(\sqrt { 6\sqrt { 6\sqrt { 6\sqrt { 6.....\infty } } } } \)
- (d)
\({ 3 }^{ \log _{ 1/3 }{ \left( \frac { 1 }{ 6 } \right) } }\)
\(\log _{ p }{ \log _{ p }{ \underbrace { \sqrt [ p ]{ \sqrt [ p ]{ \sqrt [ p ]{ ....\sqrt [ p ]{ p } } } } }_{ n\quad times } , } } \) p>0 and p\(\ne\)1, is equal to
- (a)
n
- (b)
-n
- (c)
\(\frac {1}{n}\)
- (d)
\(\log _{ 1/p }{ ({ p }^{ n }) } \)
Sum of the roots of the equation x+1=2log2(2x+3)-2log4(1980-2-x) is
- (a)
log112
- (b)
log211
- (c)
log11(0.5)
- (d)
\(\log _{ 0.5 }{ \left( \frac { 1 }{ 11 } \right) } \)
The solution of the equation 3logax+3xloga3=2 is given by
- (a)
alog3a
- (b)
(2/a)log32
- (c)
a-log32
- (d)
2-log3a
An equation \(\begin{cases} f(x)>0 \\ { f }^{ 2m }(x)=g(x) \end{cases}\) is equivalent to the system 2mlogaf(x)=logag(x), a>0, a\(\ne\)1, m∈N. The number of solutions of 2loge2x=2loge(7x-2-2x2) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite
An equation \(\begin{cases} f(x)>0 \\ { f }^{ 2m }(x)=g(x) \end{cases}\) is equivalent to the system 2mlogaf(x)=logag(x), a>0, a\(\ne\)1, m∈N. Solution set of the equation \(\log _{ ({ x }^{ 3 }+6) }{ ({ x }^{ 2 }-1) } =\log _{ ({ 2x }^{ 2 }+5x) }{ ({ x }^{ 2 }-1) } \) is
- (a)
{-2}
- (b)
{1}
- (c)
{3}
- (d)
{-2,1,3}