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<x<1, $y$ is always less than $0$ and decreases without limit as $x$ approaches zero

Only some of the above statements are correct

Answer: E

Concepts:logarithm function complex number

Difficulty rating: 1530

Small Hint:

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

Big Hint:

For $a>1,$ the real logarithm is increasing and tends to $-\infty$ as $x$ approaches $0$ from the right

Solution:

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.

1950 AMC 12 Problem 37

Answer: E

Difficulty rating: 1530

Small Hint:

Big Hint:

Solution:

Problem 37 in Other Years