Solving a Stanford University entrance exam t

Hello Friends find the value of If 5^t.5^t=200 let's have solution so, we have problem of 5^t.5^t=200 as x.x=x^2 then It will be 5^(2t)=200 divide by 25 on both sides 5^(2t)/25=200/25 25x8=200 5^(2t)/5^2=8 where 25=5^2 we know a^m/a^n=a^(m-n) by using formula, we have 5^(2t-2)=8 take 'log' on both sides log(5^(2t-2))=log(2^3) since 8=2^3 we know logm^n=nlogm (2t-2)log5=3log2 divide by 'log5' on both sides ((2t-2)log5)/log5=(3log2)/log5 where 'log5' cancels 2t-2=3log2/log5 loga/logb=logb(a) then 2t=2+3log5(2) divide by '2' on both sides 2t/2=(2+3log5(2))/2 '2' cancels we get the value of 't=(2+3log5(2))/2' verify, we have problem 5^t.5^t=200 which is same as 5^(2t)=200 put the value of 5^2(2+3log5(2))/2)=200 where '2' cancels 5^(2+3log5(2))=200 a^(m+n)=a^m.a^n then 5^2.5^(3log5(2))=200 nlogm=logm^n 25.5^(log5(2^3))=200 25.5^(log5(8))=200 we know a^loga(b)=b by using formula, we have 25.8=200 200=200 L.H.S=R.H.S which shows that the value of satisfies this equation thanks for watching this video please subscribe this channel to get the notification of my new videos ok bye