Question

Suppose that $p$ and $q$ are positive numbers for which \[\log_9 p = \log_{12} q = \log_{16} (p + q).\] what is the value of $q/p$?

Answer 1

Given
`p,q>0` and
`\log_9p=\log_(12)q=\log_(16)(p+q)=x` (say).

Then,

`image`

From the above 3 equations,

`image`

From the equations, we get

`image`

Since
`p,q>0`, the negative value is rejected.

`(q)/(p)=(1 + √(5))/(2)`