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\\ |
Revision as of 23:13, 4 April 2015
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 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.