Question

please answer the question ​

Answer 1

Answer :

${\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120} = \dfrac{3}{2}}$

Step-by-step explanation:

We have,

${ \implies\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120}}$

${ \implies\dfrac{\log(\sqrt{3})^{3} + \log (2)^{3} + \log (\sqrt {10})^{3} }{\log ( {2}^{2} \times 3 \times 10)}}$

We are aware about the log properties,

• $\boxed{ \rm\log {x}^{y} = y \log x}$
• $\boxed{ \rm\log mn = \log m + \log n}$

So, by using these properties, we get the following expressions.

${ \implies\dfrac{3\log(\sqrt{3})+ 3\log (2) +3 \log (\sqrt {10})}{\log ( {2}^{2}) + \log( 3 ) + \log (10)}}$

${ \implies\dfrac{3\log(\sqrt{3})+ 3\log (2) +3 \log (\sqrt {10})}{\log ( {2})^{2} + \log( \sqrt{3} )^{2} + \log ( \sqrt{10})^{2} }}$

${ \implies\dfrac{3\log(\sqrt{3})+ 3\log (2) +3 \log (\sqrt {10})}{2\log ( {2}) + 2 \log( \sqrt{3} ) + 2\log ( \sqrt{10}) }}$

${ \implies\dfrac{3(\log(\sqrt{3})+ \log (2) + \log (\sqrt {10}))}{2(\log ( {2}) + \log( \sqrt{3} ) +\log ( \sqrt{10})) }}$

Now, we can cancel out the same log expressions from num. and deno., By cancellation, we get the followings.

${ \implies\dfrac{3}{2}}$

Hence this is the required answer.

Answer 2

Given Question :-

Evaluate the following

${\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120}}$

$\large\underline{\sf{Solution-}}$

Consider,

$\rm :\longmapsto\:{\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120}}$

Let first evaluate numerator

Consider,

$\rm :\longmapsto\:\log\sqrt{27} + \log 8 + \log \sqrt{1000}$

We know,

$\boxed{ \tt{ \: logm \: + logn \: = \: log(mn) \: }}$

So, using this identity, we get

$\rm \:  =  \:log\bigg[ \sqrt{27} \times 8 \times \sqrt{1000} \bigg]$

$\rm \:  =  \:log\bigg[ \sqrt{3 \times 3 \times 3} \times 8 \times \sqrt{10 \times 10 \times 10} \bigg]$

$\rm \:  =  \:log\bigg[ 3 \sqrt{3} \times 8 \times 10 \sqrt{10} \bigg]$

$\rm \:  =  \:log\bigg[ 240 \sqrt{30} \bigg]$

$\rm \:  =  \:log\bigg[ 120 \times 2 \times \sqrt{30} \bigg]$

$\rm \:  =  \:log\bigg[ 120 \times \sqrt{4} \times \sqrt{30} \bigg]$

$\rm \:  =  \:log\bigg[ 120 \times \sqrt{4 \times 30} \bigg]$

$\rm \:  =  \:log\bigg[ 120 \times \sqrt{120} \bigg]$

$\rm \:  =  \:log\bigg[ 120\bigg]^{1 + \dfrac{1}{2} }$

$\rm \:  =  \:log\bigg[ 120\bigg]^{ \dfrac{2 + 1}{2} }$

$\rm \:  =  \:log\bigg[ 120\bigg]^{ \dfrac{3}{2} }$

We know,

$\boxed{ \tt{ \: log {x}^{y} = y \: logx \: }}$

So, using this identity, we get

$\rm \:  =  \:\dfrac{3}{2} \: log(120)$

So,

$\:\boxed{ \tt{ \: \log\sqrt{27} + \log 8 + \log \sqrt{1000} = \dfrac{3}{2} \: log(120) \: }}$

Now, Consider,

$\red{\rm :\longmapsto\:{\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120}}}$

$\rm \:  =  \:\dfrac{\dfrac{3}{2} \: \cancel{log120}}{ \cancel{log120}}$

$\rm \:  =  \:\dfrac{3}{2}$

Hence,

$\red{\rm :\longmapsto\:\boxed{ \tt{ \: {\dfrac{\log\sqrt{27} + \log 8 + \log \sqrt {1000}}{\log 120}}} = \frac{3}{2}}}$

More to know :-

$\boxed{ \tt{ \: log_{x}(x) = 1 \: }}$

$\boxed{ \tt{ \: log_{ {x}^{y} }( {x}^{z} ) = \frac{z}{y} \: }}$

$\boxed{ \tt{ \: {e}^{logx} = x \: }}$

$\boxed{ \tt{ \: {e}^{y \: logx} = {x}^{y} \: }}$

$\boxed{ \tt{ \: {a}^{ log_{a}(x) } = x \: }}$

$\boxed{ \tt{ \: {a}^{y log_{a}(x) } = {x}^{y} \: }}$