1950 AHSME Problems/Problem 37

Problem

If $y = \log_{a}{x}$ , $a > 1$ , which of the following statements is incorrect?

$\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ \textbf{(B)}\ \text{If }x=a,y=1 \qquad\\ \textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)} \qquad\\ \textbf{(D)}\ \text{If }0<x<a,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero} \qquad\\ \textbf{(E)}\ \text{Only some of the above statements are correct}$

Solution

Let us first check

$\textbf{(A)}\ \text{If }x=1,y=0$ . Rewriting into exponential form gives $a^0=1$ . This is certainly correct.

$\textbf{(B)}\ \text{If }x=a,y=1$ . Rewriting gives $a^1=a$ . This is also certainly correct.

$\textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)}$ . Rewriting gives $a^{\text{complex number}}=-1$ . Because $a>1$ , therefore positive, there is no real solution to $y$ , but there is imaginary.

$\textbf{(D)}\ \text{If }0<x<a,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero}$ . Rewriting: $a^y=x$ such that $x<a$ . Well, a power of $a$ can be less than $a$ only if $y<1$ . And we observe, $y$ has no lower asymptote, because it is perfectly possible to have $y$ be $-100000000$ ; in fact, the lower $y$ gets, $x$ approaches $0$ . This is also correct.

$\textbf{(E)}\ \text{Only some of the above statements are correct}$ . This is the last option, so it follows that our answer is $\boxed{\textbf{(E)}\ \text{Only some of the above statements are correct}}$

1950 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 36	Followed by Problem 38
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1950 AHSME Problems/Problem 37

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