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