Given that log_9 p=log_(12)q=log_(16)(p+q) find the value of q/p

Nico Patterson

Nico Patterson

Answered question

2022-11-04

Given that log 9 p = log 12 q = log 16 ( p + q ) find the value of q / p
This is not homework, it's just a brain teaser which I can't solve, just some hints should be sufficient, I know that from this we get:
( 1 / 4 ) log 2 ( p + q ) = ( 1 / 2 ) log 3 p = log 3 q 1 + 2 log 3 2
now I'd like to combine these quantities in some way so I can see the value of q / p from this, but I can't seem to be able to figure it out.

Answer & Explanation

Aliya Moore

Aliya Moore

Beginner2022-11-05Added 17 answers

So there is a number x such that 9 x = p, 12 x = q and 16 x = p + q
Therefore 9 x + 12 x = 16 x
One final hint: 12 = 3 × 4
Uriah Molina

Uriah Molina

Beginner2022-11-06Added 7 answers

Let L denote the common logarithmic value. Thus p = 9 L , q = 12 L , and p + q = 16 L . Let r = q / p = 12 L / 9 L = ( 4 / 3 ) L . Then
r 2 = ( 16 / 9 ) L = ( p + q ) / p = 1 + r
Can you take it from there?

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