If m ≥ n > 1, what is the largest possible value of log base m (m/n) + log base n (n/m)?

log_m(m/n) = 1 - log_m(n)

log_n(n/m) = 1 - log_n(m)
adding things up, we have

2 - log_m(n) - log_n(m)
= 2 - 1/log_n(m) - log_n(m)

if m>=n, log_n(m) >= 1

so, if you call log_n(m), that gives you

2 - 1/x - x
This achieves a maximum value of 0 at x=1, which is when m=n.