log

• J2011

  1 Jun 11, 13:43

  It's given that x and y are two real numbers such that x>1 and y>1. Find the smallest possible value of

  logx 2011 + logy 2011

  logxy 2011

   

  Thanks.

• J2011

  1 Jun 11, 13:54

  It's given that x and y are two real numbers such that x>1 and y>1. Find the smallest possible value of (log_x 2011 + log_y 2011) / log_xy 2011

   Thanks.

• eagle

  2 Jun 11, 16:35

  Hi, do a change of base to Ln and try again.

• J2011

  2 Jun 11, 20:19

   

   

   

  Hi eagle, got it. Thanks.

  (log_x 2011 + log_y 2011) / log_xy 2011

  = [(ln 2011 / ln x) + (ln 2011 / ln y)] / (ln 2011 / ln xy)

  = (ln xy)(ln x + ln y) / (ln x)(ln y)

  = (ln x + ln y)² / (ln x)(ln y)

  = 4{[(ln x + ln y) / 2] / √[(ln x)(ln y)]} ²  > 4

  Hence, the smallest possible value is 4.