Question

Find the minimum value of log_nm + log_mn, where m > 1 and n > 1.

Answer 1

When m > 1 and n > 1, then, we have 

log_nm > 0 and log_mn > 0 

Now, AM > GM for positive real numbers 

∴ \(\frac12\) (log_n m + log_m n) > (log_nm.log_mn)^\(\frac12\) ⇒ \(\frac12\) (log_n m + log_m n) ≥ \(\bigg(\frac{log_em}{log_en}.\frac{log_en}{log_em}\bigg)^{\frac12}\) 

⇒ (log_n m + log_m n) > 2 × 1 = 2 

∴ Minimum value of log_n m + log_m n is 2.