log

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.

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.

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

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.