IISER Mathematics - Logarithms And Their Properties
Exam Duration: 45 Mins Total Questions : 30
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
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
If f(x)=\(In\left( \frac { 1+x }{ 1-x } \right) ,\) then
- (a)
f(x1).f(x2)=f(x1+x2)
- (b)
f9x+2)-2f(x+1)+f(x)=0
- (c)
f(x)+f(x+1)=f(x2+x)
- (d)
\(f({ x }_{ 1 })+f({ x }_{ 2 })=f\left( \frac { { x }_{ 1 }+{ x }_{ 2 } }{ 1+{ x }_{ 1 }{ x }_{ 2 } } \right) \)
If log102,log10(2x+1), log10(2x+3) are in AP, then
- (a)
x=0
- (b)
x=1
- (c)
x=log102
- (d)
\(x=\frac { 1 }{ 2 } \log _{ 2 }{ 5 } \)
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)
\(\frac { 1 }{ \log _{ 43 }{ n! } } \)
log10tan 10+log10tan 20+....+log10tan890 is equal to
- (a)
0
- (b)
1
- (c)
27
- (d)
81
If log1227=a, then log616 is equal to
- (a)
\(2\left( \frac { 3-a }{ 3+a } \right) \)
- (b)
\(3\left( \frac { 3-a }{ 3+a } \right) \)
- (c)
\(4\left( \frac { 3-a }{ 3+a } \right) \)
- (d)
\(5\left( \frac { 3-a }{ 3+a } \right) \)
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 (4)log93+(9)log24=(10)logx83, then x is equal to
- (a)
2
- (b)
3
- (c)
10
- (d)
30
If x,y,z are in GP and ax=by=cz, then
- (a)
logba=logcb
- (b)
logcb=logac
- (c)
logac=logba
- (d)
logcb=2logac
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)
\({ a }^{ \frac { 1 }{ 2-\log _{ a }{ z } } }\quad \)
The value of 3log45-5log43 is
- (a)
0
- (b)
1
- (c)
2
- (d)
none of these
If x18=y21=z28, then 3,3logyx, 3logzy,7logxz are in
- (a)
AP
- (b)
GP
- (c)
HP
- (d)
AGP
If \(\frac { 1 }{ \log _{ 3 }{ \pi } } +\frac { 1 }{ \log _{ 4 }{ \pi } } \)>x, then x be
- (a)
2
- (b)
3
- (c)
\(\pi\)
- (d)
none of these
If log3{5+4log3(x-1)}=2, then x is equal to
- (a)
2
- (b)
4
- (c)
8
- (d)
log216
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
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)
\(\frac { 1 }{ \alpha \beta \gamma \delta } \)
\(\log _{ p }{ \log _{ p }{ \underbrace { \sqrt [ p ]{ \sqrt [ p ]{ \sqrt [ p ]{ ....\sqrt [ p ]{ p } } } } }_{ n\quad times } , } } \) p>0 and p\(\ne\)1, is equal to
- (a)
n
- (b)
-n
- (c)
\(\frac {1}{n}\)
- (d)
\(\log _{ 1/p }{ ({ p }^{ n }) } \)
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
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 In 2x=2In (4x-15) is
- (a)
0
- (b)
1
- (c)
2
- (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}
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