### Mathematics - Logarithms And Their Properties

#### Question - 1

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 $\frac { 1+2ac }{ 2c-abc-1 }$

#### Question - 2

If log5120+(x-3)-2log5(1-5x-3)

• A 1
• B 2
• C 3
• D 4

#### Question - 3

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

#### Question - 4

The number log203 lies in

• A $\left( \frac { 1 }{ 4 } ,\frac { 1 }{ 3 } \right)$
• B $\left( \frac { 1 }{ 3 } ,\frac { 1 }{ 2 } \right)$
• C $\left( \frac { 1 }{ 2 } ,\frac { 3 }{ 4 } \right)$
• D $\left( \frac { 3 }{ 4 } ,\frac { 4 }{ 5 } \right)$

#### Question - 5

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

#### Question - 6

If $y={ 2 }^{ \frac { 1 }{ \log _{ x }{ 4 } } }$ , then x is equal to

• A $\sqrt { y }$
• B y
• C y2
• D y3

#### Question - 7

if log2x+log2y$\ge$6, then the least value of x+y is

• A 4
• B 8
• C 16
• D 32

#### Question - 8

A rational number which is 50 times its own logarithm to the base 10 is

• A 1
• B 10
• C 100
• D 1000

#### Question - 9

If x=log5(1000) and y=log7(2058), then

• A x>y
• B x<y
• C x=y
• D none of these

#### Question - 10

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 $f({ x }_{ 1 })+f({ x }_{ 2 })=f\left( \frac { { x }_{ 1 }+{ x }_{ 2 } }{ 1+{ x }_{ 1 }{ x }_{ 2 } } \right)$