1997 AMC 12 Problem 17

Attempt Problem 17 of the 1997 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 1997 AMC 12 solutions, or check the answer key.

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17.

A line

x=k

intersects the graph of

y=\log_5x

and the graph of

y=\log_5(x+4).

The distance between the points of intersection is

0.5.

Given that

k=a+\sqrt b,

where

a

and

b

are integers, what is

a+b?

6

7

8

9

10

Because both points have

x

-coordinate

k

, their distance is the difference of their logarithms

Combine the logarithms and exponentiate base

5

The vertical distance is

\begin{aligned} \log_5(k+4)-\log_5k &=\frac12,\\ \frac{k+4}{k}&=\sqrt5. \end{aligned}

Thus

\frac{k+4}{k}=\sqrt5,

k=\frac{4}{\sqrt5-1}=1+\sqrt5.

Therefore

a+b=1+5=6,

and A is correct.

Problem 17 in other years