1961 AMC 12 Problem 19

Attempt Problem 19 of the 1961 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 1961 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

19.

Consider the graphs of

y=2\log x

and

y=\log2x.

We may say that:

they do not intersect

they intersect at

1

point only

they intersect at

2

points only

they intersect at a finite number of points but greater than

2

they coincide

Use

2\log x=\log(x^2)

for

x>0

Equate the logarithm arguments and respect the logarithm’s domain

On the domain

x>0,

equality requires

\log(x^2)=\log(2x),

x^2=2x.

The only positive solution is

x=2,

giving exactly one intersection.

Therefore, the correct answer is B.

Problem 19 in other years