Difference between revisions of "2019 AIME II Problems/Problem 6"
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-Stormersyle | -Stormersyle | ||
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+ | ==Solution 5(Substitution)== | ||
+ | Let <math>y = \log _{b} x</math> | ||
+ | Then we have | ||
+ | <math>3\log _{b} (y\sqrt{x}) = 56</math> | ||
+ | <math>\log _{y} x = 54</math> | ||
+ | <math>y^{54} = x</math> | ||
+ | <math>3\log _{b} (y * y^{27}) = 3\log _{b} y^{28} = 56</math> | ||
+ | <math>\log _{b} y = \dfrac{2}{3}</math> | ||
+ | <math>b^{2/3} = y</math> | ||
+ | <math>b^{36} = x</math> | ||
+ | <math>y = \log _{b} x = 36</math> | ||
+ | <math>b = 36^{3/2} = \fbox{216}</math> | ||
==See Also== | ==See Also== | ||
{{AIME box|year=2019|n=II|num-b=5|num-a=7}} | {{AIME box|year=2019|n=II|num-b=5|num-a=7}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 23:51, 8 May 2019
Contents
Problem 6
In a Martian civilization, all logarithms whose bases are not specified as assumed to be base , for some fixed
. A Martian student writes down
and finds that this system of equations has a single real number solution
. Find
.
Solution 1
Using change of base on the second equation to base b,
Substituting this into the
of the first equation,
We can manipulate this equation to be able to substitute a couple more times:
However, since we found that ,
is also equal to
. Equating these,
Solution 2
We start by simplifying the first equation to
Next, we simplify the second equation to
Substituting this into the first equation gives
Plugging this into
gives
-ktong
Solution 3 (Elegant)
Apply change of base to to yield:
which can be rearranged as:
Apply log properties to
to yield:
Substituting
into the equation
yields:
So
Substituting this back in to
yields
So,
-Ghazt2002
Solution 4 (Easiest)
From the first equation we have that , so
. From the second equation we have that
, so now set
and
. Substituting, we have that
, so
. We also have that
, so
. This means that
, so
, and
.
-Stormersyle
Solution 5(Substitution)
Let
Then we have
See Also
2019 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 5 |
Followed by Problem 7 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.