Question

if log(x+y/3)=1/2(logx+logy),then find the value of x/y+y/x

Answer 1

given that 
log (x+y/3) = 1/2 (logx+logy)
log (x+y/3) = 1/2 (log xy)
log (x+y/3) = log √xy
remove the log on the both side 
(x+y/3) = √xy
(x+y/3)² = xy
x²+y²+2xy/9 = xy
x²+y²+2xy = 9xy
x²+y² = 9xy-2xy
x²+y² = 7xy   ------------------------(1)
form the another 
= x/y +y/x
= x²+y²/xy               from (1)
= 7xy / xy
= 7

Answer 2

Step-by-step explanation:

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

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

log x+y/3=logxy1/2

x+y/3=✓xy

x+y=3✓xy

squaring on both sides

(x+y)2=3✓xy2

x2+y2+2xy=9xy

x2+y2+2xy=9xy

x2+y2+2xy=9xy

x2+y2=7xy

Divide on both sides

x2+y2/x+y=7xyx2/xy+y2/xy=7xy/xy

x/y+y/x=7