Question

Given that log x = m + n and log y = m - n, express the value of log ` ( 10x  ) / ( y ^ 2  )`  in terms of m and n.

Answer 1

 Given that
log x = m + n ; 
log y = m - n ;
Consider the expression log ` ( 10x )/( y^2 ) ` : 

log ` ( 10x  ) / ( y^2  ) `

= log 10 x - log y² 
⇒ log 10 x - 2 log y  ... [ n log_a m = log_a mⁿ]
⇒ log 10 + log x - 2 log y ...[ log_a m + log_a n = log_a mn ]
⇒ 1 +  log x - 2 log y   
⇒  1 + m + n - 2 ( m - n )
⇒  1 + m + n - 2m + 2n
⇒ log ` ( 10x  )/( y^2 )` = 1 - m + 3n.