Question

If x2+y2=10xy prove that 2 log( x + y)=logx+logy+2 log2 + log3​

Answer 1

Solution :

• Given : x² + y² = 10xy
• To prove : 2log(x + y) = logx + logy + 2log2 + log3

Proof :

We have ;

x² + y² = 10xy

Now ,

Adding 2xy both the sides , we get ;

=> x² + y² + 2xy = 10xy + 2xy

=> (x + y)² = 12xy

=> (x + y)² = 2²·3xy

Now ,

Taking log both the sides , we get ;

=> log[(x+y)²] = log[2²·3xy]

=> 2log(x+y) = log(2²) + log3 + logx + logy

=> 2log(x+y) = 2log2 + log3 + logx + logy

=> 2log(x+y) = logx + logy + 2log2 + log3

Hence proved .

Properties used :

• log(ab) = loga + logb

• log(a/b) = loga - logb

• logaⁿ = n·loga

Answer 2

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