Difference between revisions of "2018 AMC 12A Problems/Problem 14"
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+ | ==Solution 4== | ||
+ | <math>\log_{3x} 4=\log_{2x} 8</math> is the same as <math>2\log_{3x} 2=3\log_{2x} 2</math> | ||
+ | |||
+ | Using Reciprocal law, we get <math>\log_{(3x)^\frac{1}{2}} 2=\log_{(2x)^\frac{1}{3}} 2</math> | ||
+ | |||
+ | <math>\Rightarrow (3x)^\frac{1}{2}=(2x)^\frac{1}{3}</math> <math>\Rightarrow 27x^3=4x^2</math> <math>\Rightarrow \frac{x^3}{x^2}=\frac{4}{27}=x</math> | ||
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+ | <math>\therefore \frac{p}{q}=\frac{4}{27}</math> <math>\Rightarrow p+q=4+27=</math> <math>\boxed{\textbf{(D) } 31}</math> | ||
==See Also== | ==See Also== | ||
{{AMC12 box|year=2018|ab=A|num-b=13|num-a=15}} | {{AMC12 box|year=2018|ab=A|num-b=13|num-a=15}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 04:22, 25 January 2019
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
Base switch to log 2 and you have .
Then . so and we have leading to (jeremylu)
Solution 2
If you multiply both sides by
then it should come out to * = *
that then becomes * = *
which simplifies to
so now = putting in exponent form gets
=
so =
dividing yields and
- Pikachu13307
Solution 3
We can convert both and into and , respectively, giving:
Converting the bases of the right side, we get
Dividing both sides by , we get
Which simplifies to
Log expansion allows us to see that
, which then simplifies to
Thus,
And
-lepetitmoulin
Solution 4
is the same as
Using Reciprocal law, we get
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 13 |
Followed by Problem 15 |
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 | |
All AMC 12 Problems and Solutions |
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