1950 AMC 12 Problem 37

Question

If

y=\log_a x,

a>1,

which of the following statements is incorrect?

If

x=1,

y=0

If

x=a,

y=1

If

x=-1,

y

is imaginary (complex)

If

0\lt x\lt1,

y

is always less than

0

and decreases without limit as

x

approaches zero

Only some of the above statements are correct

Check choices A and B directly from the definition of a logarithm

For

a>1,

the real logarithm is increasing and tends to

-\infty

as

x

approaches

0

from the right

We have

\log_a1=0

and

\log_aa=1.

A real logarithm is not defined at

-1,

though its complex values are nonreal. For

a>1,

\log_a x<0

on

0<x<1,

and it tends to

-\infty

as

x

approaches

0

from the right. Thus statements A through D are all correct, making the claim that only some are correct the incorrect statement.

Thus, the correct answer is E.

Problem 37 in other years