Question

If xsquare+ysquare=30xy,then prove that 2log(x+y)=logx+logy+5log2

Answer 1

Step-by-step explanation:

On the RHS of the equation instead of 30xy, there should be 32 xy to give the required proof: $\\ \\ \because 32=2^5$

So, if it is 32xy on the RHS, then:

${x}^{2} + {y}^{2} = 32xy \\ applying \: log \: on \: both \: sides \: we \: \\ find: \\ log \: {x}^{2} + log \: {y}^{2} = log \: (32 \: x \: y) \\ \therefore \: 2log \: x + 2log \: y = log \: x + log \: y + log \: 32 \\ 2( log \: x + log \: y ) = log \: x + log \: y + log \: {2}^{5} \\ 2 \: log \: (x y) = log \: x + log \: y +5 log \: {2}\\ \\ Logarithmic \: laws\: used\: are\:\\ given\: below\\ log (mn)= log m+ log n\: or\: vice-versa\\ log m^n= n logm$