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Difference between revisions of "1950 AHSME Problems/Problem 37"

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

Revision as of 16:08, 15 March 2017

Problem

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

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

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 $\boxed{\textbf{(E)}\ \text{Only some of the above statements are correct}}$

See Also

1950 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 36 		Followed by
Problem 38
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50
All AHSME Problems and Solutions

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