for a positive constant a find dy/dx , where y = a^{t + 1/t} and x = (t + 1/t)^a.in the solution of this ques when we find out dy/dt i dont understand loga?

we have,
y = a^t+1/t
taking log both sides we get
log(y) = log(a^t+1/t)
log(y) = (t+1/t)log(a)         log(a^m) = mlog(a)
now differentiating both sides w.r.t. t we get
1/y . dy/dt = log(a)(1-1/t²)
dy/dt = a^t+1/t[log(a)(1-1/t²)]
Now,
x = (t + 1/t)^a
log(x) = log(t + 1/t)^a
log(x) =(a)log(t + 1/t)
diff. w.r.t. t
1/x . dx/dt = (a)[1/(t+1/t)](1-1/t²)
dx/dt = (t + 1/t)^a-1.(a).(1-1/t²)

dy/dx = a^t+1/tlog(a)/[(t + 1/t)^a-1.(a)]