Question

If x²+y²=25xy, then prove that 2 log (x+y)=3log3+logx+logy​

Answer 1

Step-by-step explanation:

Given :-

x²+y²=25xy

To find :-

Prove that 2 log (x+y)=3log3+logx+logy.

Solution :-

Given that :

x²+y²=25xy

On adding '2xy' both sides then

=> x²+y²+2xy =25xy+2xy

=> x²+y²+2xy =(25+2)xy

=>x²+y²+2xy =27xy

=> x²+2xy+y² = 27xy

LHS is in the form of a²+2ab+b²

Where, a = x and b=y

We know that

(a+b)² = a²+2ab+b²

=> (x+y)² = 27xy

On taking 'logarithms' both sides then

=> log (x+y)² = log 27xy

=> log (x+y)² = log (27×x×y)

=> log (x+y)² = log (3×3×3×x×y)

=>log (x+y)² = log (3³×x×y)

We know that

log a^m = m log a

=> 2 log (x+y) = log (3³×x×y)

We know that

log (ab) = log a + log b

=> 2 log (x+y) = log 3³ + log x + log y

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

Since log a^m = m log a

Hence, Proved.

Answer:-

If x²+y²=25xy then 2 log(x+y) = 3log3+logx+logy

Used formulae:-

• (a+b)² = a²+2ab+b²

• log a^m = m log a

• log (ab) = log a + log b