2018 AMC 12A Problems/Problem 14
Contents
Problem
The solutions to the equation , where
is a positive real number other than
or
, can be written as
where
and
are relatively prime positive integers. What is
?
Solution 1
We apply the Change of Base Formula, then rearrange:
By the logarithmic identity
it follows that
from which the answer is
~jeremylu (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 2
We will apply the following logarithmic identity:
which can be proven by the Change of Base Formula:
We rewrite the original equation as
from which
Therefore, the answer is
~MRENTHUSIASM
Solution 3
By the logarithmic identity the original equation becomes
By the logarithmic identity
we multiply both sides by
then apply the Change of Base Formula to the left side:
Therefore, the answer is
~Pikachu13307 (Fundamental Logic)
~MRENTHUSIASM (Reconstruction)
Solution 4
We can convert both and
into
and
respectively:
Converting the bases of the right side, we get
Dividing both sides by
we get
from which
Expanding this equation gives
Thus, we have
from which the answer is
~lepetitmoulin (Solution)
~MRENTHUSIASM (Reformatting)
Solution 5 (Exponential Form)
Let We convert the equations to the exponential form:
Cubing the first equation and squaring the second equation, we have
Applying the Transitive Property, we get
from which the answer is
~MRENTHUSIASM
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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All AMC 12 Problems and Solutions |
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