Tag: properties of logarithms

Questions Related to properties of logarithms

The value of $\,3^{\textstyle \log _4\,5}\,+\,4^{\textstyle \log _5\,3}\,-5^{\textstyle \log _4\,3}\,-3^{\textstyle \log _5\,4}$ is equal to

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

  1. $\,\,0$

  2. $\,\,1$

  3. $\,\,2$

  4. $3$

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Correct Option: A
Explanation:
Use $\,a^{\textstyle \log _b\,c}\,=\,c^{\textstyle \log _b\,a}$
So, $[5^{\textstyle \log _4\,3}=3^{\textstyle \log _4\,5}]$
and $[4^{\textstyle \log _5\,3}=3^{\textstyle \log _5\,4}]$
$\Rightarrow\,\,3^{\textstyle \log _4\,5}\,+\,4^{\textstyle \log _5\,3}\,-\,3^{\textstyle \log _4\,5}\,-\,4^{\textstyle \log _5\,3}\,=\,0$

Ans: A

$\log _{25} 25$ is equal to

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

  1. $1$

  2. $0$

  3. $\infty$

  4. none of these

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Correct Option: A
Explanation:

We know $log _a  a = 1.         \therefore log _{25} 25 = 1$

Find the value of $x$ which satisfies $4.18^{x} = 36.54$.

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

  1. $0.86$

  2. $1.43$

  3. $1.80$

  4. $2.17$

  5. $2.52$

Show answer
Correct Option: E
Explanation:

Given, ${4.18}^{x} = 36.54$


$\Rightarrow x = \log _{ 4.18 }{ 36.54 } $

$\Rightarrow x=\dfrac { \log { 36.54 }  }{ \log { 4.18 }  } =2.52$

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?

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

  1. $4.5$

  2. $9$

  3. $18$

  4. $27$

Show answer
Correct Option: B
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

$\log 8 {64}$ is equal to____

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

  1. $2$

  2. $3$

  3. $4$

  4. $5$

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Correct Option: A
Explanation:

$\log _8(64)$

$=\log _8(8^2)$
$=2\log _88$
$=2$                     $\because(\log _aa=1)$
Hence, A is the correct option.

$\log 4{64}$ is equal to___

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

  1. $2$

  2. $3$

  3. $4$

  4. $5$

Show answer
Correct Option: B
Explanation:

$\log _4(64)$

$=\log _4(4^3)$
$=3\log _44$
$=3$                              $\because(\log _aa=1)$
Hence, B is the correct option.

$\log 2 {64}$ is equal to___

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

  1. $2$

  2. $3$

  3. $4$

  4. $6$

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Correct Option: D
Explanation:

$\log _2(64)$

$=\log _2(2^6)$
$=6\log _22$
$=6$                        $\because(\log _aa=1)$
Hence, D is the correct option.

$\log _{\sqrt{3}} {81}$ is equal to

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

  1. $8$

  2. $4$

  3. $5$

  4. $6$

Show answer
Correct Option: A
Explanation:

$\log _{\sqrt3}81$

$=\cfrac1{\frac12}\log _381$
$=2\log _33^4$
$=2\times4\log _33$
$=2\times4$
$=8$
Hence, A is the correct option.

$\log _\sqrt{5} {625}$ is equal to

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

  1. $8$

  2. $4$

  3. $5$

  4. $6$

Show answer
Correct Option: A
Explanation:

$\log _{\sqrt5}625$

$=\cfrac1{\frac12}\log _5625$
$=2\log _55^4$
$=2\times4\log _55$
$=2\times4$
$=8$
Hence, A is the correct option.

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

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

  1. 1

  2. -1

  3. 0

  4. 2

Show answer
Correct Option: C
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.