Question

If x² + y²=loxy, then prove that 2 log (x+4)=log x + log y+2 log 2 + log 3​

Answer 1

Step-by-step explanation:

Given :-

x²+y² = 10xy

Correction :-

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

To find :-

Prove that 2 log (x+y)=log x + log y

+2 log 2 + log 3

Solution :-

Given that

x²+y² = 10xy

On adding 2xy both sides then

=> x²+y² +2xy = 10xy +2xy

=>x²+ 2xy + y² = 12xy

=> (x+y)² = 12xy

On taking logarithms both sides then

=> log (x+y)² = log (12xy)

We know that

log a^m = m log a

=> 2 log (x+y) = log (12xy)

=> 2 log (x+y) = log (2×2×3×x×y)

=> 2 log (x+y) = log (2²×3×x×y)

We know that

log (ab) = log a + log b

=> 2 log(x+y)=log 2² + log 3 + log x + log y

We know that

log a^m = m log a

=>2 log(x+y)=2 log 2 + log 3 + log x + log y

=>2 log(x+y) = log x + log y 2 log 2 + log 3

Hence, Proved.

Answer:-

If x²+y² = 10xy then 2 log (x+y)

= log x + log y 2 log 2 + log 3

Used formulae:-

→ log a^m = m log a

→ log (ab) = log a + log b

→ (a+b)² = a²+2ab+b²