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)
\(\frac { 1+2ac }{ 2c-abc-1 } \)
If log5120+(x-3)-2log5(1-5x-3)
- (a)
1
- (b)
2
- (c)
3
- (d)
4
Ifxn>xn-1>....x2>x1>1, then the value of logx1logx2logx3....logxn\(\log _{ xn }{ { x }_{ n }^{ { x }_{ n-1 }^{ { . }^{ { .x1 }^{ } } } } } \) is
- (a)
0
- (b)
1
- (c)
2
- (d)
undefined
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 log2x+log2y\(\ge\)6, then the least value of x+y is
- (a)
4
- (b)
8
- (c)
16
- (d)
32
If x=log5(1000) and y=log7(2058), then
- (a)
x>y
- (b)
x<y
- (c)
x=y
- (d)
none of these
For y=logax to be defined 'a' must be
- (a)
any positive real number
- (b)
any number
- (c)
\(\ge e\)
- (d)
any positive real number \(\ne\)1
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)
\(\frac { 1 }{ \log _{ 43 }{ n! } } \)
log10tan 10+log10tan 20+....+log10tan890 is equal to
- (a)
0
- (b)
1
- (c)
27
- (d)
81
If (4)log93+(9)log24=(10)logx83, then x is equal to
- (a)
2
- (b)
3
- (c)
10
- (d)
30
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 log0.3(x-1)
- (a)
\((-\infty,1)\)
- (b)
(1,2)
- (c)
\((2,\infty)\)
- (d)
none of these
If x18=y21=z28, then 3,3logyx, 3logzy,7logxz are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
AGP
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
The least value of expression 2log10x-logx0.01 is
- (a)
2
- (b)
4
- (c)
6
- (d)
8
The value of {logba.logcb.logdc.logad} is
- (a)
0
- (b)
log abcd
- (c)
log 1
- (d)
1
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)
\(\log _{ 2 }{ \left( \frac { 1 }{ 256 } \right) } \)
If \(\frac { \log { a } }{ (b-c) } =\frac { \log { b } }{ (c-a) } =\frac { \log { c } }{ (a-b) } ,\) then ab+c.bc+a.ca+b is equal to
- (a)
0
- (b)
1
- (c)
a+b+c
- (d)
logba.logcb.logac
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)
\({ 3 }^{ \log _{ 1/3 }{ \left( \frac { 1 }{ 6 } \right) } }\)
If \(\log _{ a }{ x } =\alpha ,\log _{ b }{ x } =\beta ,\log _{ c }{ x } =\gamma \quad and\quad \log _{ d }{ x } =\delta ,\quad x\neq 1\quad and\quad a,b,c,d\neq 0,>q,\quad then\quad \log _{ abcd }{ x } \)equals
- (a)
\(\le \frac { \alpha +\beta +\gamma +\delta }{ 16 } \)
- (b)
\(\ge \frac { \alpha +\beta +\gamma +\delta }{ 16 } \)
- (c)
\(\frac { 1 }{ { \alpha }^{ -1 }+{ \beta }^{ -1 }+{ \gamma }^{ -1 }+{ \delta }^{ -1 } } \)
- (d)
\(\frac { 1 }{ \alpha \beta \gamma \delta } \)
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)
\(\log _{ 0.5 }{ \left( \frac { 1 }{ 11 } \right) } \)
If ax=b, by=c, cz=a, x=logbak1,y=logcbk2,z=logack3, then k1k2k3 is equal to
- (a)
1
- (b)
abc
- (c)
(xyz)
- (d)
0
If \(\frac { \ln { a } }{ b-c } =\frac { \ln { b } }{ c-a } =\frac { \ln { c } }{ a-b } ,\) a,b,c>0, then
- (a)
ab+c.bc+a.ca+b=1
- (b)
ab+c.bc+a.ca+b ≥3
- (c)
ab+c.bc+a.ca+b=3
- (d)
ab+c.bc+a.ca+b ≥ 3(3)1/3
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. 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}
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 \(\log { (x-9) } +2\log { \sqrt { (2x-1) } } =2\) is
- (a)
\(\left\{ \phi \right\} \)
- (b)
{1}
- (c)
{2}
- (d)
{13}
Equations of the form (i) f(logax)=0, a>0, a\(\ne\)1 and (ii) g(logxA)=0, A>0, then Eq. (i) is equivalent to f(t)=0, where t=logax. If t1,t2,t3,....,tk are the roots of f(t)=0, then logax=t2,...,logax=tk and Eq. (ii) is equivalent to f(y)=0, where y=logxA. If y1,y2,y3,....,yk are the roots of f(y)=0, then logxA=y1, logxA=y2,....,logxA=yk. The number of solutions of the equation \(\frac { 1-2{ \left( \log { { x }^{ 2 } } \right) }^{ 2 } }{ \log { { x } } -2{ \left( \log { { x }^{ 2 } } \right) }^{ 2 } } =1\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
infinite
Equations of the form (i) f(logax)=0, a>0, a\(\ne\)1 and (ii) g(logxA)=0, A>0, then Eq. (i) is equivalent to f(t)=0, where t=logax. If t1,t2,t3,....,tk are the roots of f(t)=0, then logax=t2,...,logax=tk and Eq. (ii) is equivalent to f(y)=0, where y=logxA. If y1,y2,y3,....,yk are the roots of f(y)=0, then logxA=y1, logxA=y2,....,logxA=yk. The number of solutions of the equation \(\log { _{ x }^{ 3 } } 10-6\log { _{ x }^{ 2 } } 10+11\log _{ x }{ 10 } -6=0\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Equations of the form (i) f(logax)=0, a>0, a\(\ne\)1 and (ii) g(logxA)=0, A>0, then Eq. (i) is equivalent to f(t)=0, where t=logax. If t1,t2,t3,....,tk are the roots of f(t)=0, then logax=t2,...,logax=tk and Eq. (ii) is equivalent to f(y)=0, where y=logxA. If y1,y2,y3,....,yk are the roots of f(y)=0, then logxA=y1, logxA=y2,....,logxA=yk. If \(\frac { { (In\quad x) }^{ 2 }-3Inx+3 }{ Inx-1 } <1,\) then x belongs to
- (a)
(0,e)
- (b)
(1,e)
- (c)
(1,2e)
- (d)
(0,3e)