Question

If `log_12 27 = a`, then what is `log_6 16`?

Answer 1

A good problem

Given

`a=log_12 27=log_(12)3^3=3log_(12)3`
`=>a=3/log_3 12=3/log_3(3xx2^2)`

`=>a=3/(log_3 3+2log_3 2`

`=>a=3/(1+2log_3 2)`

`=>(1+2log_3 2)=3/a`

`=>log_3 2=1/2(3/a-1)=(3-a)/(2a)`

`=>log_2 3=(2a)/(3-a)`

Now `log_6 16=log_6 2^4=4log_6 2=4/log_2 6`

`=4/log_2 (2xx3)=4/(log_2 2+log_2 3`

`=4/(1+log_2 3)=4/(1+(2a)/(3-a)`

`=(4(3-a))/(3-a+2a)=(12-4a)/(3+a)`