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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.
Multiple choice logarithm and its uses basic mathematical concepts physics

If $x=\log _{ a }{ bc } ,y=\log _{ b }{ ca } ,z=\log _{ c }{ ab } $, then the value of $\dfrac { 1 }{ 1+x } +\dfrac { 1 }{ 1+y } +\dfrac { 1 }{ 1+z } $ will be

  1. $x+y+z$

  2. $1$

  3. $ab+bc+ca$

  4. $abc$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Now, $1+x=\log _{ a }{ a } +\log _{ a }{ bc } =\log _{ a }{ abc } $
$\Rightarrow \dfrac { 1 }{ 1+x } =\log _{ abc }{ a } $
Similarly, $\dfrac { 1 }{ 1+y } =\log _{ abc }{ b } $ and $\dfrac { 1 }{ 1+z } =\log _{ abc }{ c } $
$\therefore \dfrac { 1 }{ 1+x } +\dfrac { 1 }{ 1+y } +\dfrac { 1 }{ 1+z } $
                    $=\log _{ abc }{ a } +\log _{ abc }{ b } +\log _{ abc }{ c } $
                    $=\log _{ abc }{ abc } =1$

Multiple choice logarithm and its uses basic mathematical concepts physics

The characteristic of a number having $m$ $(m>1)$ digits is given by,

  1. $m-1$

  2. $m+1$

  3. $m$

  4. None of the above
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If a number is $N>0$, then $\log _{10}N$ will have two parts, the integral part is known the characteristic and the decimal part is known as mantissa.

$2$ digit belongs from $[10,100]$ where $\log _{10}10=1$ and $\log _{10}100=2$
Similarly, $m$ digit number belongs from $[10^{m-1},10^m]$, where $\log _{10}10^{m-1}=m-1$ and $\log _{10}10^m=m$.
Thus any number between $[10^{m-1},10^m]$ will have $m-1$ as the integral part.
Thus the characteristic of a number having $m$ digits is given by $m-1$.

Multiple choice logarithm and its uses basic mathematical concepts physics

Calculate $x$, to the nearest tenth: $\log _{12} 640 = x$

  1. $1.7$

  2. $2.6$

  3. $2.8$

  4. $53.3$

  5. $7,680$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given, $x= \log _{ 12 }{ 640 } $

$=\log _{ 12 }{ 144\times 4.44 } $
$=\log _{ 12 }{ 144 } +\log _{ 12 }{ 4.44 } $
$=2+\log _{ 12 }{ 4.44 }$
$= 2.6$