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Quizzes (598) Questions (13441)

Questions Related to physics

Multiple choice logarithm and its uses basic mathematical concepts physics

If $\log _{10}e=0.4343$, then $\log _{10}1016$ is

  1. $2.99$

  2. $3$

  3. $3.006949$

  4. $3.02$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\log _{10}1016\Rightarrow \dfrac{\log 1016}{\log 10}$

$\Rightarrow \dfrac{3.006949}{1}$
$\rightarrow$ Option $C$ is correct

Multiple choice logarithm and its uses basic mathematical concepts physics

Multiple Correct:

Which of the following statements are true

  1. $\log _{ 2 }{ 3 } <\log _{ 12 }{ 10 } $

  2. $\log _{ 6 }{ 5 } <\log _{ 7 }{ 8 } $

  3. $\log _{ 3 }{ 26 } <\log _{ 2 }{ 9 } $

  4. $\log _{ 16 }{ 15 } >\log _{ 10 }{ 11 } >\log _{ 7 }{ 6 }$
Reveal answer Fill a bubble to check yourself
B,C Correct answer
Multiple choice logarithm and its uses basic mathematical concepts physics

The solution of the equation $\log _{7}\log _{5}(\sqrt {x^{2}}+5+x)=0$

  1. $x=2$

  2. $x=3$

  3. $x=0$

  4. $x=-2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\log _{7}\log _{5}(\sqrt{x^{2}}+5+x)=0$

$\log _{5}(\sqrt{x^{2}}+5+x)=1$
$\sqrt{x^{2}+5+x=5}$
$\sqrt{x^{2}+x=0}$
$x^{2}+x^{2}+2\sqrt[x]{x^{2}}=0$
$x=0$


Multiple choice logarithm and its uses basic mathematical concepts physics

The value of $\displaystyle\sum _{r=1}^{n}log\left ( \dfrac{a^{r}}{b^{r-1}} \right )$ is

  1. $\dfrac{n}{2}log\left ( \dfrac{a^{n}}{b^{n}} \right )$

  2. $\dfrac{n}{2}log\left ( \dfrac{a^{n}}{b^{n+1}} \right )$

  3. $\dfrac{n}{2}log\left ( \dfrac{a^{n+1}}{b^{n+1}} \right )$

  4. $\dfrac{n}{2}log\left ( \dfrac{a^{n+1}}{b^{n-1}} \right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Now,
$\displaystyle\sum _{r=1}^{n}\log\left ( \dfrac{a^{r}}{b^{r-1}} \right )$
$=\displaystyle\sum _{r=1}^{n}\left(\log a^{r}-\log b^{r-1}\right)$
$=\displaystyle\sum _{r=1}^{n}\left(r\log a-(r-1)\log b\right)$
$=(\log a)\times \dfrac{n(n+1)}{2}-\log b\times\dfrac{(n-1)n}{2}$
$=\dfrac{n}{2}\left(\log a^{n+1}-\log b^{n-1}\right)$
$=\dfrac{n}{2}\log\left(\dfrac{a^{n+1}}{b^{n-1}}\right)$
Multiple choice logarithm and its uses basic mathematical concepts physics

Find the value of $\log _{10}{\left(0.\bar{9}\right)}$

  1. $0$

  2. $1$

  3. $-1$

  4. $2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
To find value of $\log _{10}{0.\bar9}$

Let $x=0.\bar{9}=0.999999...$

$\Rightarrow 10x=9.99999....$

$\Rightarrow 10x-x=9$

$\Rightarrow 9x=9$

$\Rightarrow x=\dfrac{9}{9}=1$

$\therefore x=1$

Let $y=\log _{10}{1}$

$\Rightarrow 1={10}^{y}$

$\Rightarrow {10}^{y}={10}^{0}$                 (since ${10}^{0}=1$)

Since bases are same we can equate the powers

$\therefore y=0$

Hence, $\log _{10}{\left(0.\bar{9}\right)}=0$

Multiple choice logarithm and its uses basic mathematical concepts physics

If $y=a\log\left|x\right|+bx^{2}+x$ has extreme values at $x=2$ and $x=-4/3$ then 

  1. $a=12,b=-10$

  2. $a=4,b=-3/4$

  3. $a=-6,b=1/4$

  4. $none$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The derivative y' = a/x + 2bx + 1. Setting y' = 0 at x = 2 and x = -4/3 gives a system of equations: a/2 + 4b + 1 = 0 and a/(-4/3) - 8b/3 + 1 = 0. Solving these yields a=12 and b=-10.

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,

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

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

$=5050$