Difference between revisions of "1950 AHSME Problems/Problem 37"
| Line 1: | Line 1: | ||
| − | == | + | ==Problem== |
If <math> y \equal{} \log_{a}{x}</math>, <math> a > 1</math>, which of the following statements is incorrect? | If <math> y \equal{} \log_{a}{x}</math>, <math> a > 1</math>, which of the following statements is incorrect? | ||
| Line 6: | Line 6: | ||
\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<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}</math> | \textbf{(E)}\ \text{Only some of the above statements are correct}</math> | ||
==Solution== | ==Solution== | ||
| − | {{solution}} | + | Let us first check |
| + | |||
| + | <math>\textbf{(A)}\ \text{If }x=1,y=0</math>. Rewriting into exponential form gives <math>a^0=1</math>. This is certainly correct. | ||
| + | |||
| + | <math>\textbf{(B)}\ \text{If }x=a,y=1</math>. Rewriting gives <math>a^1=a</math>. This is also certainly correct. | ||
| + | |||
| + | <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 solution to <math>y</math>. Is it possible to have complex powers? Let's use process of elimination to find out the right answer. | ||
| + | |||
| + | <math>\textbf{(D)}\ \text{If }0<x<a,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>. Now, we need to use logic to see if this is true or not. Assume that this is true; but that would mean that <math>\textbf{(C)}</math> and <math>\textbf{(E)}</math> would both be answers. This is definitely not possible, so it follows that <math>\textbf{(E)}</math> is incorrect, therefore the answer is <math>\boxed{\textbf{(E)}\ \text{Only some of the above statements are correct}}</math> | ||
==See Also== | ==See Also== | ||
Revision as of 19:15, 10 April 2013
Problem
If $y \equal{} \log_{a}{x}$ (Error compiling LaTeX. Unknown error_msg), 
, 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 solution to 
. Is it possible to have complex powers? Let's use process of elimination to find out the right answer.
. 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.
. Now, we need to use logic to see if this is true or not. Assume that this is true; but that would mean that 
and 
would both be answers. This is definitely not possible, so it follows that is incorrect, therefore the answer is 
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 | ||
