Tag: basic mathematical concepts

Questions Related to basic mathematical concepts

The greatest value of $(4\log _{10}{x}-\log _{2}{(0.0001)})$ for $0 < x < 1$ is

logarithm and its uses basic mathematical concepts physics

  1. $4$

  2. $-4$

  3. $8$

  4. $-8$

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Correct Option: A

If $P$ is the number of natural numbers whose logarithm to the base $10$ have the characteristic $p$ and $Q$ is the number of natural numbers logarithm of whose reciprocals to the base $10$ have the characteristic $-q$, then find the value of $\log _{10}P-\log _{10}Q$.

logarithm and its uses basic mathematical concepts physics

  1. $p-q+1$

  2. $p+q-1$

  3. $p+q$

  4. $p-q$

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Correct Option: A
Explanation:

Let $x$ and $y$ be the numbers whose logarithm to the base $10$ have the characteristic $p$ and $q$ respectively.
$10^{p}\leq x< 10^{p+1}\Rightarrow P=10^{p+1}-10^{p}\Rightarrow P=9\times 10^{p}$
Similarly, $10^{q-1}< y\leq 10^{q}$
$\Rightarrow $   $Q=10^{q}-10^{q-1}=10^{q-1}\left ( 10-1 \right )=9\times 10^{q-1}$
$\therefore $   $\log _{10}P-\log _{10}Q=\log _{10}\left ( P/Q \right )=\log _{10}10^{p-q+1}=p-q+1$

Find the number of positive integers which have the characteristic $3$, when the base of the logarithm is $7$.

logarithm and its uses basic mathematical concepts physics

  1. $2058$

  2. $1029$

  3. $1030$

  4. $2060$

Show answer
Correct Option: A
Explanation:

Let there be $N$ integers whose characteristic is 3, when base of log is 7
Then, $\log _{ 7 }{ N } =x$ where $3\le x<4$
As $3\le x<4$
$3\le \log _{ 7 }{ N } <4\ \Rightarrow { 7 }^{ 3 }\le N<{ 7 }^{ 4 }\ \Rightarrow N={ 7 }^{ 4 }-{ 7 }^{ 3 }\ \Rightarrow N=2401-343=2058$ 

The value of $\displaystyle anti\log _5\left [\frac {\tan^2\left (\frac {\pi}{5}\right )+\tan^2\left (\frac {2\pi}{5}\right )+20}{\cot^2\left (\frac {\pi}{5}\right )+\cot^2\left (\frac {2\pi}{5}\right )+28}\right ]$ is equal to

logarithm and its uses basic mathematical concepts physics

  1. odd number

  2. even number

  3. prime number

  4. composite number

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Correct Option: A,C
Explanation:
$\frac { { tan }^{ 2 }(\frac { \Pi  }{ 5 } )+{ tan }^{ 2 }(\frac { 2\Pi  }{ 5 } )+20 }{ { cot }^{ 2 }(\frac { \Pi  }{ 5 } )+{ cot }^{ 2 }(\frac { 2\Pi  }{ 5 } )+28 } \\ =\frac { { (\sqrt { 5-2\sqrt { 5 }  } ) }^{ 2 }+(\sqrt { \frac { 5 }{ 5-2\sqrt { 5 }  }  } )^{ 2 }+20 }{ { (\frac { 1 }{ \sqrt { 5-2\sqrt { 5 }  }  } ) }^{ 2 }+{ (\sqrt { \frac { 5-2\sqrt { 5 }  }{ 5 } ) }  }^{ 2 }+28 } \\ =\frac { 5-2\sqrt { 5 } +\frac { 5 }{ 5-2\sqrt { 5 }  } +20 }{ \frac { 1 }{ 5-2\sqrt { 5 }  } +\frac { 5-2\sqrt { 5 }  }{ 5 } +28 } \\ =\frac { 5[(5-{ 2\sqrt { 5 } ) }^{ 2 }+5+20(5-2\sqrt { 5 } )] }{ 5+(5-{ 2\sqrt { 5 } ) }^{ 2 }+140(5-2\sqrt { 5 } ) } \\ =\frac { 5(45-20\sqrt { 5 } +5+100-40\sqrt { 5 } ) }{ 5+45-20\sqrt { 5 } +700-200\sqrt { 5 }  } \\ =\frac { 750-300\sqrt { 5 }  }{ 750-300\sqrt { 5 }  } \\ antilog _{ 5 }[\frac { { tan }^{ 2 }(\frac { \Pi  }{ 5 } +{ tan }^{ 2 }(\frac { 2\Pi  }{ 5 } )+20 }{ { cot }^{ 2 }(\frac { \Pi  }{ 5 } )+{ cot }^{ 2 }(\frac { 2\Pi  }{ 5 } )+28 } ]={ 5 }^{ 1 }\quad =5$

Evaluate using logarithm table: $\dfrac {28.45 \times \sqrt [3] {0.3254}}{32.43 \times \sqrt [5] {0.3046}}$

logarithm and its uses basic mathematical concepts physics

  1. $0.7666$

  2. $0.7656$

  3. $0.5686$

  4. $0.2936$

Show answer
Correct Option: B
Explanation:

Let $y=\dfrac { 28.45\times \sqrt [ 3 ]{ .3254 }  }{ 32.43\times \sqrt [ 3 ]{ .3046 }  } $

$ \ln { y } =\ln { 28.45 } +\ln { \sqrt [ 3 ]{ .3254 }  } -(\ln { 32.43 } +\ln { \sqrt [ 5 ]{ .3046 }  } )\ \ln { y } =\ln { 25.45 } +\dfrac { 1 }{ 3 } \ln { .3245- } \ln { 32.43 } -\dfrac { 1 }{ 5 } \ln { .4046 } \ \ln { y } =3.236+(-.375)-3.479-(-.237)\ \ln { y } =-.381$
$ y=$ anti $\ln { (-.381) } $

$ y=.7656$
So, option B is correct.

If $\log _{10} 2 = 0.3010$, then the number of digits in $2^{64}$ is

logarithm and its uses basic mathematical concepts physics

  1. $18$

  2. $24$

  3. $22$

  4. $20$

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Correct Option: D
Explanation:

Given $\log _{ 10 }{ 2 } =0.301$

$\log _{ 10 }{ 2^{64} } =64 \times \log _{ 10 }{ 2 } =64 \times 0.3010=19.264$
$\Rightarrow 2^{64}=10^{19.264}$
The number of digits in $10^{19}$ is $20$ , there will be $21$ digits from $10^{21}$
The number $10^{19.264}$ lies between them
Therefore the number of digits in $10^{19.264}$ is $20$
Therefore the correct option is $D$

If $\log _{10} 3 = 0.4771$, then the number of zeros after the decimal in $3^{-100}$ is

logarithm and its uses basic mathematical concepts physics

  1. $47$

  2. $48$

  3. $49$

  4. $50$

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Correct Option: A
Explanation:

The number of zeroes will be
$=|log _{10}(3^{-100})|$
$=|-100(log _{10}(3))|$
$=|-47.71|$
$=47.71$
Taking the integral part (since number of zeroes has to be an integer, there will be $47$ zeros.

Approximate of $\log _{11}21$ is

logarithm and its uses basic mathematical concepts physics

  1. 1.27

  2. 1.21

  3. 1.18

  4. 1.15

  5. 1.02

Show answer
Correct Option: A
Explanation:

Approximate value of $\log _{ 11 }{ 21 } $

$=\log _{ 11 }{ (7\times 3) } $
$=\log _{ 11 }{ 7 } +\log _{ 11 }{ 3 }$
$ =0.8115+0.4581$
$=1.27$