Let a and b be real numbers greater than 1 for which there exists a positive real number c, different from 1, such that `2(log_a c +log_b c)=9log_ab c`. Find the largest possible value of `log_a b`.

Answer

2

Explanation

`a gt 1, b gt 1`
`2(log_(a)c + log_(b)c)=9 log_(ab)c`
`rArr 2[log c[(log b+log a)/(log a log b)]]=0 (log c)/(log a + log b)`
`rArr 2(log a+ log b)^(2) = 9(log a)(log b)`
`rArr 2(log a)^(2)+2(log b)^(2)+4(log a)(log b)= 9(log a)(log b)`
`rArr 2log_(b)a + 2log_(a)b=5`
`rArr t+(1)/(t)=(5)/(2)`, where `t=log_(a)b`
`rArr 2r^(2)-5t+2=0`
`rArr (2t-1)(t-2)=0`
`rArr t = 1//2, t=2`
`rArr log_(a)b=1//2` or `log_(a)b=2`