If `log_(4)A=log_(6)B=log_(9)(A+B)," then "[4(B//A)]("where "[*]` represents the greatest integer function ) equals _______.

Answer

6

Explanation

Let ` log_(4)0A=log_(6)B=log_(9)(A+B) = x`
`rArr A = 4^(x), B=6^(x) and A+B=9^(x)`
` A+B = 9^(x) rArr 4^(x) + 6^(x) = 9^(x)`
` rArr 2^(2x)+2^(x)*3^(x)=3^(2x)`
`rArr (3/2)^(2x)-(3/2)^(x) -1=0`
` rArr(3/2)^(x) = (1+sqrt5)/2`
` rArr B/A = (3/2)^(x) = (1+sqrt5)/2`
` rArr 4B/A = 4((1+sqrt5)/2)`
` rArr [4B/A] = 6`