Solving a 'Stanford' University entrance exam | t=?

Added: 2026-05-09

[00:00:00]Hello Friends find the value of 't' If 9^t.9^t=90 let's have a solution so, we have a problem of 9^t.9^t=90 as we know x.x=x^2 then It can be written as (9^t)^2=90 (a^m)^n=a^mn then 9^(2t)=90 It can be solved by taking 'log' on both sides log9^(2t)=log90 logm^n=nlogm we have 2tlog9=log90 divide by 'log9' on both sides (2tlog9)/log9=log90/log9 log9 cancels 2t=log(3x30)/log9 since 90=3x30 logmn=logm+logn 2t=(log3+log30)/log9 separate it into fractions 2t=log3/log9+log30/log9 as 30=6x5 so 2t=log3/log9+log(6x5)/log9 2t=log3/log9+(log6+log5)/log9 next, we have 2t=log3/log9+log6/log9+log5/log9 as 9=3x3=3^2 and 6=3x2 then 2t=log3/log3^2+log(3x2)/log3^2+log5/log3^2 2t=log3/2log3+(log3+log2)/2log3+log5/2log3 (1/2) common 2t=1/2(1+log3/log3+log2/log3+log5/log3) log3 cancels multiply by '2' on both sides 2x2t=2x1/2(2+log2/log3+log5/log3) 2 cancels 4t=2+log2/log3+log5/log3 loga/logb=logb(a) then 4t=2+log3(2)+log3(5) I hope you understood divide by '4' on both sides 4 cancels then we get the value of 't' t=(2+log3(2)+log3(5))/4 in the next step, I'm going to verify 9^t.9^t=90 as by solving 9^t.9^t=90 we get 9^(2t)=90 so we can write here 9^(2t)=90 put the value of 't' 9^2((2+log3(2)+log3(5)/4)=90 2.2=4 9^((2+log3(2)+log3(5)/2)=90 9=3.3=3^2 3^2((2+log3(2)+log3(5)/2)=90 2 cancels we get 3^(2+log3(2)+log3(5))=90 a^(m+n)=a^m.a^n then It will be 3^2.3^log3(2).3^log3(5)=90 a^loga(b)=b using this formula we have 9.2.5=90 90=90 you can see L.H.S=R.H.S which shows that the value of t=(2+log3(2)+log3(5))/4 satisfies this equation of 9^t.9^t=90 thanks for watching this video please subscribe this channel to get the notification of my new videos ok bye