Question

If a2 + b2 = 47ab

log (a +b/2) = 1/2 (log a + log b)

Answer 1

a^2+b^2=47ab
add both sides 2ab
a^2+b^2+2ab=47ab+2ab
(a+b)^2=49ab
(a+b)^2=7^2ab
[(a+b)/7]^2=ab
take both sides log
log[(a+b)/7]^2=log (ab)
[here we use log x^n= nlog x
log(xy)=log x +log y]
2log (a+b)/7= log a+log b
log(a+b)/7=1/2[loga +logb]

Answer 2

That will be log (a + b/7)

a² + b² = (a + b)² - 2ab

(a + b)² -  2ab = 47ab
(a + b)² = 49ab
a + b = 7√ab
(a + b)/7 = √ab

Now multiply log to both sides ,
log [(a + b)/7] = log √ab
log [(a + b)/7] = 1/2 log ab
log [(a + b)/7 ] = 1/2 (log a + log b)      (∵ logₐ mn = logₐm + logₐn)

Hence Proved

Hope This Helps You!