Mathematics · Quantitative Aptitude

Logarithms

246 Questions

Logarithms are mathematical operations that determine the exponent required for a base to reach a specific number. This topic tests the application of logarithmic properties, changing bases, and solving complex equations. It is a high-yield topic for quantitative aptitude in competitive exams.

Logarithmic expressionsBase change propertiesSolving log equationsInfinite series logsCharacteristic values

Logarithms Questions

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

If $\left( \log _{ 3 }{ x }  \right) \left( \log _{ x }{ 2x }  \right) \left( \log _{ 2x }{ y }  \right) =\log _{ x }{ { x }^{ 2 } } $, then what is $y$ equal to?

  1. $4.5$
  2. $9$
  3. $18$
  4. $27$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$(\log _{ 3 }{ x }) (\log _{ x }{ 2x }) (\log _{ 2x }y)=\log _{x} x^{2}$ .... $(i)$
$(\log _{ 3 }{ x }) (\log _{ x }{ 2x }) (\log _{ 2x }y)$
$=\left(\dfrac { \log x}{\log 3}\right) \left(\dfrac {\log 2x}{\log x} \right) \left(\dfrac { \log y}{\log 2x}\right)$ .......[using base change formula]
$=\dfrac { \log { y }  }{ \log { 3 }  } =\log _{ 3 }{ y } $
Also, $\log _{ x }{ { x }^{ 2 } } =2\log _{ x }{ x } =2$
So, $\log _{ 3 }{ y } =2$ .... From $(i)$
$\Rightarrow y={ 3 }^{ 2 }=9$
Hence, option B is correct

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

If $\displaystyle \log _{ 2a }{ a } =x$, $\log _{ 3a }{  2a } =y$ and $\log _{ 4a }{  3a } =z$, then $xyz-2yz$ is equal to

  1. 1

  2. -1

  3. 0

  4. 2
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We have,

${{\log } _{2a}}a=x,{{\log } _{3a}}2a=y,{{\log } _{4a}}3a=z$

$ {{\log } _{2a}}a=x $

$ \dfrac{\log a}{\log 2a}=x $

Similarly,

$ {{\log } _{3a}}2a=y $

$ \dfrac{\log 2a}{\log 3a}=y $

Similarly,

$ {{\log } _{4a}}3a=z $

$ \dfrac{\log 3a}{\log 4a}=z $


Therefore,,

$ =xyz-2yz $

$ =\dfrac{\log a}{\log 2a}\times \dfrac{\log 2a}{\log 3a}\times \dfrac{\log 3a}{\log 4a}-2\times \dfrac{\log 2a}{\log 3a}\times \dfrac{\log 3a}{\log 4a} $

$ =\dfrac{\log a}{\log 4a}-2\times \dfrac{\log 2a}{\log 4a} $

$ =\log a-\log 4-\log a-2\times \left( \log 2a-\log 4a \right) $

 $ =-\log 4-2\times \left( \log 2a-\log 2a-\log 2 \right) $

 $ =-\log 4+2\log 2 $

 $ =-\log 4+\log 4 $

 $ =0 $

So, the value is 0.

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

The set of solutions for the equation $\log _{ 10 }{ \left( { a }^{ 2 }-15a \right)  } =2$ consists of:

  1. Two integers

  2. One integer and one fraction

  3. two irrational numbers

  4. two non-real numbers

  5. no numbers, that is, the set is empty
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$\log _{10}{(a^{2}-15a)}=2$

$\Rightarrow a^{2}-15a=10^{2}$
$\Rightarrow a^{2}-15a-100=0$
Discrimnant$= (15)^{2}-(4)(-100)=225+400= 625>0$
$(a-20)(a+5)=0$
$a=20,-5$
(Two integers)

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

If $\log _{2}x + \log _{4}x + \log _{64} x = 5$, then the value of $x$ will be

  1. $8$
  2. $16$
  3. $7$
  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ \log _{ 2 }{ x } +\log _{ 4 }{ x } +\log _{ 64 }{ x } = 5$


We know that,
 
$ \log _{ b }{ a }=\dfrac { \log _{c} { a }  }{  \log _{c} {  b }   } $

$\therefore \dfrac { \log { x }  }{  \log {  2 }   } + \dfrac { \log {  x }  }{  \log {  4 }   } +\dfrac { \log {  x }  }{  \log {  64 } }  = 5$


 Assuming base $2$ for numerator and denominator, we get

$\dfrac { \log x }{ 1 } +\dfrac { \log x }{ 2 }  +\dfrac { \log x }{ 6 }  = 5$

$ 6\log x + 3\log x+\log x = 30$

$ 10 \log x = 30$

$ \log _{ 2 }{ { x }^{ 10 } }  =30 \ \ \ \ \dots (\log _{  { y}  }x^n=n \log _{ y }x)$

$ \\ { x }^{ 10 } = { 2 }^{ 30 }$

So, $x = { 2 }^{ 3 }=8$

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

The solution of the equation ${ 4 }^{ \log _{ 2 }{ \log { x }  }  }=\log { x } -{ \left( \log { x }  \right)  }^{ 2 }+1$ is

  1. $x=1$
  2. $x=4$
  3. $x=e$
  4. $x={e}^{2}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ \log _{ 2 }{ \log { x }  }  }$ is meaningful if $x>1$
Since ${ 4 }^{ \log _{ 2 }{ \log { x }  }  }={ \left( { 2 }^{ \log _{ 2 }{ \log { x }  }  } \right)  }^{ 2 }={ \left( \log { x }  \right)  }^{ 2 }$         .......$\left[\because  a^{ \log _{ a }{ x }  }=x,a>0,a\neq 1 \right] $     

So the given equation {${ 4 }^{ \log _{ 2 }{ \log { x }  }  }=\log { x } -{ \left( \log { x }  \right)  }^{ 2 }+1$} reduces to
$2{ \left( \log { x }  \right)  }^{ 2 }-\log { x } -1=0$
$\Rightarrow \log { x } =1,\log { x } =-\dfrac12$
But for $x>1$, $\log { x } >0$
 so $\log { x } =1$ i.e., $x=e$

Ans: C

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

If $\log _{ 4 }{ \left( x \right)  } =12\quad $, then $\log _{ 2 }{ \left( x/4 \right)  } $ is equal to

  1. $11$
  2. $48$
  3. $-12$
  4. $22$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

If $\log _{ 4 }{ \left( x \right)  } =12$ then,


$x={ 4 }^{ 12 }\$ 

$\frac { x }{ 4 } ={ 4 }^{ 11 }$

Hence, 

$\log _{ 2 }{ \left( \frac { x }{ 4 }  \right)  } =\log _{ 2 }{ \left( { 4 }^{ 11 } \right)  } \$ 

                 $=\log _{ 2 }{ \left( \left( { 2 }^{ 2 } \right) ^{ 11 } \right)  } \$ 

                 $=\log _{ 2 }{ \left( { 2 }^{ 22 } \right)  }$ 

                 $ =22$

Hence option D is correct.

Multiple choice maths logarithms more about logarithms a relation between logarithmic functions properties of logarithms

If $a, b, c$ are positive numbers such that $a^{\log _37}=27, b^{\log _711}=49, c^{\log _{11}25}=\sqrt{11}$, then the sum of digits of $S=a^{(\log _37)^2}+b^{(\log _711)^2}+c^{(\log _{11}25)^2}$ is

  1. 15

  2. 17

  3. 19

  4. 21
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

${ a }^{ \log _{ 3 }{ 7 }  }=27,\quad { b }^{ \log _{ 7 }{ 11 }  }=49,\quad { c }^{ \log _{ 11 }{ 25 }  }=\sqrt { 11 } $
$S=({ a }^{ \log _{ 3 }{ 7 } })^{\log _{ 3 }{ 7 }  }+{ \left( { b }^{ \log _{ 7 }{ 11 }  } \right)  }^{ \log _{ 7 }{ 11 }  }+{ \left( { c }^{ \log _{ 11 }{ 25 }  } \right)  }^{ \log _{ 11 }{ 25 }  }$
$={ \left( 27 \right)  }^{ \log _{ 3 }{ 7 }  }+{ \left( 49 \right)  }^{ \log _{ 7 }{ 11 }  }+{ \left( \sqrt { 11 }  \right)  }^{ \log _{ 11 }{ 25 }  }$
$={ 3 }^{ 3\log _{ 3 }{ 7 }  }+{ 7 }^{ 2\log _{ 7 }{ 11 }  }+{ 11 }^{ { 1 }/{ 2 }\log _{ 11 }{ 25 }  }$
$={ 7 }^{ 3 }+{ 11 }^{ 2 }+{ 25 }^{ { 1 }/{ 2 } }$
$=469$
Sum of digits $=19$
Hence, C is correct.