Question

Can you solve this exercise? please solve so I understand

Answer 1

Answer:

Answer is 2). 4

Explanation:

${ \rm{(2 - log_{ \sqrt{2} }10)(2 - log_{ \sqrt{5} }10)}}$

• let's open the brackets:

${ \rm{(2 \times 2) + (2 \times - log_{ \sqrt{5} }10 ) + (2 \times - log_{ \sqrt{2} }10) + ( log_{ \sqrt{2} }10 \times log_{ \sqrt{5} }10) }} \\ \\ = { \rm{4 - 2 log_{ \sqrt{5} }10 - 2 log_{ \sqrt{2} }10 + ( log_{ \sqrt{5} }10)( log_{ \sqrt{2} }10)}}$

• change log√5 to log√2:

${ \rm{4 - 2 log_{ \sqrt{2} }10 - (\frac{2 log_{ \sqrt{2} }10 }{ log_{ \sqrt{2} } \sqrt{5} } ) + ( log_{ \sqrt{2} }10)( \frac{ log_{ \sqrt{2} }10 }{ log_{ \sqrt{2} } \sqrt{5} } )}} \\ \\ = { \rm{4 - 2 log_{ \sqrt{2} }10 - ( \frac{4 log_{ \sqrt{2} }10}{ log_{ \sqrt{2} } \sqrt{5} }) }} \\ \\ = { \rm{4 - 2 log_{ \sqrt{2} }10 - ( \frac{8 log_{ \sqrt{2} }(5 \times 2) }{ log_{ \sqrt{2} }5} )}} \\ \\ = { \rm{4 - 2 log_{ \sqrt{2} }(5 \times 2) - \frac{8 log_{ \sqrt{2} }2}{ log_{ \sqrt{2} }5 } }}$

• now let's change log√2 to log10

$= { \rm{4 - \frac{2 log_{10}10}{ log_{10} \sqrt{2} } - \frac{8 log_{10}2}{ log_{10} \sqrt{2} } \div \frac{ log_{10}5 }{ log_{10} \sqrt{2} } }} \\ \\ = { \rm{4 - \frac{2}{0.3} - \frac{2.41}{0.3} \times \frac{0.3}{0.7} }} \\ \\ = { \rm{4 + \frac{41}{70} }} \\ \\ { \rm{ = 4 \frac{41}{70} \: \approx \: 4 }}$

Answer 2

$= 4 - 4 \: log_{5}(10) - 4 log_{2}(10) + 4 log_{2}(10) log_{5}(10) \\ \\ or \: \\ \\ = 4$

the answer is 2

hope it helps

see the attachment for explanation