1988 AMC 12 Problem 26

Attempt Problem 26 of the 1988 AMC 12 below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 1988 AMC 12 solutions, or check the answer key.

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26.

Suppose that

p

and

q

are positive numbers for which

\begin{aligned} \log_9(p)&=\log_{12}(q)\\ &=\log_{16}(p+q). \end{aligned}

What is the value of

\frac{q}{p}?

\frac43

\frac12(1+\sqrt3)

\frac85

\frac12(1+\sqrt5)

\frac{16}{9}

Answer: D

Concepts:logarithms exponential equations quadratic equations

Difficulty rating: 2360

Small hint:

Call the common logarithm value

t

Big hint:

If

r=\frac{q}{p},

compare

(\frac{16}{9})^t

with

(\frac{12}{9})^t

Solution:

Let the common value be

t.

Then

p=9^t,\ q=12^t,

and

p+q=16^t.

Put

r=\frac{q}{p}=(\frac{4}{3})^t.

Dividing the last equation by

p

gives

\begin{aligned} 1+r&=(\frac{16}{9})^t\\ &=((\frac{4}{3})^t)^2=r^2. \end{aligned}

Since

r>0,

r=\frac{1+\sqrt5}{2}.

Thus the correct answer is D.