Difference between revisions of "1950 AHSME Problems/Problem 37"
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==Problem== | ==Problem== | ||
− | If <math> y | + | If <math> y = \log_{a}{x}</math>, <math> a > 1</math>, which of the following statements is incorrect? |
<math>\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ | <math>\textbf{(A)}\ \text{If }x=1,y=0 \qquad\\ | ||
\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\\ | ||
− | \textbf{(D)}\ \text{If }0<x< | + | \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> | ||
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<math>\textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)}</math>. Rewriting gives <math>a^{\text{complex number}}=-1</math>. Because <math>a>1</math>, therefore positive, there is no real solution to <math>y</math>, but there is imaginary. | <math>\textbf{(C)}\ \text{If }x=-1,y\text{ is imaginary (complex)}</math>. Rewriting gives <math>a^{\text{complex number}}=-1</math>. Because <math>a>1</math>, therefore positive, there is no real solution to <math>y</math>, but there is imaginary. | ||
− | <math>\textbf{(D)}\ \text{If }0<x< | + | <math>\textbf{(D)}\ \text{If }0<x<1,y\text{ is always less than 0 and decreases without limit as }x\text{ approaches zero}</math>. Rewriting: <math>a^y=x</math> such that <math>x<a</math>. Well, a power of <math>a</math> can be less than <math>a</math> only if <math>y<1</math>. And we observe, <math>y</math> has no lower asymptote, because it is perfectly possible to have <math>y</math> be <math>-100000000</math>; in fact, the lower <math>y</math> gets, <math>x</math> approaches <math>0</math>. This is also correct. |
− | <math>\textbf{(E)}\ \text{Only some of the above statements are correct}</math>. This is the last option, so | + | <math>\textbf{(E)}\ \text{Only some of the above statements are correct}</math>. This is the last option, so it follows that our answer is <math>\boxed{\textbf{(E)}\ \text{Only some of the above statements are correct}}</math> |
==See Also== | ==See Also== |
Latest revision as of 17:09, 15 March 2017
Problem
If , , which of the following statements is incorrect?
Solution
Let us first check
. Rewriting into exponential form gives . This is certainly correct.
. Rewriting gives . This is also certainly correct.
. Rewriting gives . Because , therefore positive, there is no real solution to , but there is imaginary.
. Rewriting: such that . Well, a power of can be less than only if . And we observe, has no lower asymptote, because it is perfectly possible to have be ; in fact, the lower gets, approaches . This is also correct.
. This is the last option, so it follows that our answer is
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 36 |
Followed by Problem 38 | |
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All AHSME Problems and Solutions |
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