Question

if x / y + y/x = 7 show that log (x+y) /3 = 1/2 log x + 1/2 log y​

Answer 1

x/y + y/x = 7

Taking LCM

( x² + y² ) / xy = 7

x² + y² = 7xy

Adding 2xy on both sides

x² + y² + 2xy = 9xy

Since x² + y² + 2xy = ( x + y)²

( x + y)² = 9xy

( x + y)² / 9 = xy

( x + y)² / 3² = xy

Using laws of exponents a^m / b^m = ( a/b)^m

[ ( x + y) / 3 ]² = xy

Taking log on both sides

log [ ( x + y) / 3 ]² = log xy

Using Power rule log a^m = mlog a

2log ( x + y) / 3 = log xy

log ( x + y)/3 = 1/2 log xy

Using Product rule log ab = log a + log b

log ( x + y)/3 = 1/2 ( log x + log y)

log ( x + y)/3 = 1/2 log x + 1/2 log y

Hence shown