Tag: logarithm and its uses

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Multiple choice logarithm and its uses basic mathematical concepts physics

Let $A=\dfrac{1}{6}((\log _{2}{3}))^{3}-(\log _{2}{6}))^{3}-(\log _{2}{12}))^{3}+(\log _{2}{24}))^{3})$. Then the value of $2^{A}$ is :

  1. $72$

  2. $70$

  3. $68$

  4. $None\ of \ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Using log properties and the identity (x-y)^3 = x^3 - 3x^2y + 3xy^2 - y^3, the expression simplifies to a value that results in 2^A = 72.

Multiple choice logarithm and its uses basic mathematical concepts physics

If $x=500,y=100$ and $z=5050$, then the value of $(\log _{ xyz }{ { x }^{ z } } )(1+\log _{ x }{ yz } )$ is equal to.

  1. 500

  2. 100

  3. 5050

  4. 10
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Given,

$\left(\log _{xyz}\left(x^z\right)\right)\left(1+\log _x\left(yz\right)\right)$

$\left(\log _{xyz}\left(x^z\right)\right)\left(1+\log _x\left(yz\right)\right)$

$=z\log _{xzy}\left(x\right)\left(\log _x\left(zy\right)+1\right)$

from given w have,

$=5050\log _{(500 \times 5050 \times 100)}\left(500\right)\left(\log _{500}\left((5050 \times 100)\right)+1\right)$

$=\dfrac{5050\log _e \left(505000\right)}{\log _e \left(252500000\right)}+5050\log _{252500000}\left(500\right)$

$=5050$

Multiple choice logarithm and its uses basic mathematical concepts physics

Find the mantissa of the logarithm of the number $0.002359$.

  1. $3710$

  2. $3718$

  3. $3728$

  4. $3742$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

We have to find $\log (0.002359)$

Firstly, we will write $0.002359$ in standard form.
So, $0.002359 = 2.359 \times 10^{-3}$
Here, characteristic is -3.

To find the mantissa of $\log (0.002359)$, we first look in the row starting with 23. In this row, look at the number in the column headed by 5. The number is 3711.

Now, move to the column of mean differences and look under the column headed by 9 in the row corresponding to 23. We see the number 17 here.

Add this number to 3711. We get the number 3728. This is the required mantissa of $\log (0.002359)$.

Mantissa of $\log 23.598$, $\log 2.3598$ and 0.023598 is the same (only characteristics are different).

Multiple choice logarithm and its uses basic mathematical concepts physics

The domain of the function $f(x)=[log _{10}(\frac{5x-x^2}{4})]^{{1}/{2}}$ is 

  1. $- \infty < x < \infty $

  2. $1\le x \le 4$

  3. $4\le x \le 16$

  4. $-1\le x \le 1$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$log _{10}\dfrac{5x-x^2}{4}>0$
$\dfrac{5x-x^2}{4}\geq 1$
$5x-x^2\geq 4$
$x^2-5x+4 \leq 0$
$(x-4)(x-1) \leq 0$
$x $ belongs to $[1,4]$
So the domain is $1 \leq x \leq 4$
Multiple choice logarithm and its uses basic mathematical concepts physics

The value of $\displaystyle \log _{\frac{1}{20}}40$ is

  1. greater than zero.

  2. smaller than zero.

  3. greater than zero and smaller than one.

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Let,  $y=\displaystyle \log _{\frac{1}{20}}40$
$\Rightarrow y=-\displaystyle \log _{20}40 \quad [\because \log _{1/a}b=-\log _ab]$
Now $20 < 40<20^2\Rightarrow \log _{20}20<\log _{20}40<\log _{20}20^2$
$\Rightarrow 1< \log _{20}40<2 \quad [\because \log a^m=m\log a, \log _aa=1]$
$\Rightarrow 1<-y<2\Rightarrow -2<y<-1$

Multiple choice logarithm and its uses basic mathematical concepts physics

The value of $\displaystyle \log _{\frac{2}{3}}\frac{5}{6}$ is

  1. less than zero.

  2. greater than zero and less than one.

  3. greater than one.

  4. none of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\displaystyle \frac { 2 }{ 3 } <\frac { 5 }{ 6 } <1$
Taking log with base $\displaystyle \frac { 2 }{ 3 } $
$\displaystyle \log _{ \frac { 2 }{ 3 }  }{ \frac { 2 }{ 3 }  } >\log _{ \frac { 2 }{ 3 }  }{ \frac { 5 }{ 6 }  } >\log _{ \frac { 2 }{ 3 }  }{ 1 } $  (Since the base $\displaystyle \frac { 2 }{ 3 } <1)$
$\displaystyle \Rightarrow 1>\log _{ \frac { 2 }{ 3 }  }{ \frac { 5 }{ 6 }  } >0$

Multiple choice logarithm and its uses basic mathematical concepts physics

Value of $\displaystyle \log _{4}18 $ is:

  1. an irrational number

  2. a rational number

  3. natural number

  4. whole number
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\log _{4}{18}=\log _{2^{2}}{(2.3^{2})}=\dfrac{1}{2}\log _{2}(2.3^{2})$
$=\dfrac{1}{2}\left [ \log _{2}{2}+\log _{2}{3^{2}} \right ]=\dfrac{1}{2}\left [ 1+2.\log _{2}{3} \right ]$

$\log _{2}{3}$ is an irrational number.

Hence, $\dfrac{1}{2}\left [ 1+2.\log _{2}{3} \right ]$ is also an irrational number.