Difference between revisions of "1950 AHSME Problems/Problem 42"
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==Solution== | ==Solution== | ||
+ | ==Solution 1:== | ||
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Taking the log, we get <math>\log_x 2 = x^{x^{x^{.^{.^.}}}}=2 </math>, and <math>\log_x 2 = 2</math>. Solving for x, we get <math>2=x^2</math>, and <math>\sqrt{2}=x \Rightarrow \mathrm{(D)}</math> | Taking the log, we get <math>\log_x 2 = x^{x^{x^{.^{.^.}}}}=2 </math>, and <math>\log_x 2 = 2</math>. Solving for x, we get <math>2=x^2</math>, and <math>\sqrt{2}=x \Rightarrow \mathrm{(D)}</math> | ||
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+ | ==Solution 2:== | ||
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+ | <math>x^{x^{x^{.^{.^.}}}}=2</math> is the original equation. If we let <math>y=x^{x^{x^{.^{.^.}}}}</math>, then the equation can be written as <math>y=2</math>. This also means that <math>x^y=2</math>, considering that adding one <math>x</math> to the start and then taking that <math>x</math> to the power of <math>y</math> does not have an effect on the equation, since <math>y</math> is infinitely long in terms of <math>x</math> raised to itself forever. It is already known that <math>y=2</math> from what we first started with, so this shows that <math>x^y=x^2=2</math>. If <math>x^2=2</math>, then that means that <math>\sqrt{2}=x \Rightarrow \mathrm{(D)}</math>. | ||
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+ | This is just a faster way once you get used to it, instead of taking a log of the function. | ||
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+ | ~mathmagical | ||
==See Also== | ==See Also== | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 01:21, 14 January 2023
Problem
The equation is satisfied when
is equal to:
Solution
Solution 1:
Taking the log, we get , and
. Solving for x, we get
, and
Solution 2:
is the original equation. If we let
, then the equation can be written as
. This also means that
, considering that adding one
to the start and then taking that
to the power of
does not have an effect on the equation, since
is infinitely long in terms of
raised to itself forever. It is already known that
from what we first started with, so this shows that
. If
, then that means that
.
This is just a faster way once you get used to it, instead of taking a log of the function.
~mathmagical
See Also
1950 AHSC (Problems • Answer Key • Resources) | ||
Preceded by Problem 41 |
Followed by Problem 43 | |
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All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.