Difference between revisions of "2020 AIME II Problems/Problem 3"
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==Solution 2== | ==Solution 2== | ||
− | Because <math>\log_a{b^c}=c\log_a{b},</math> we have that <math>20\log_{2^x} 3 = 2020\log_{2^{x+3}} 3,</math> or <math>\log_{2^x} 3 = 101\log_{2^{x+3}} 3.</math> Since <math>\log_a{b}=\dfrac{1}{\log_b{a},</math> <math>\log_{2^x} 3=\dfrac{1}{\log_{3} 2^x},</math> and <math>101\log_{2^{x+3}} 3=101\dfrac{1}{\log_{3}2^{x+3}},</math> thus resulting in <math>\log_{3}2^{x+3}=101\log_{3} 2^x,</math> or <math>\log_{3}2^{x+3}=\log_{3} 2^{101x}.</math> We remove the base 3 logarithm and the power of 2 to yield <math>x+3=101x,</math> or <math>x=\dfrac{3}{100}.</math> | + | Because <math>\log_a{b^c}=c\log_a{b},</math> we have that <math>20\log_{2^x} 3 = 2020\log_{2^{x+3}} 3,</math> or <math>\log_{2^x} 3 = 101\log_{2^{x+3}} 3.</math> Since <math>\log_a{b}=\dfrac{1}{\log_b{a}},</math> <math>\log_{2^x} 3=\dfrac{1}{\log_{3} 2^x},</math> and <math>101\log_{2^{x+3}} 3=101\dfrac{1}{\log_{3}2^{x+3}},</math> thus resulting in <math>\log_{3}2^{x+3}=101\log_{3} 2^x,</math> or <math>\log_{3}2^{x+3}=\log_{3} 2^{101x}.</math> We remove the base 3 logarithm and the power of 2 to yield <math>x+3=101x,</math> or <math>x=\dfrac{3}{100}.</math> |
Our answer is <math>\boxed{3+100=103}.</math> | Our answer is <math>\boxed{3+100=103}.</math> | ||
+ | |||
==Solution 3 (Official MAA)== | ==Solution 3 (Official MAA)== | ||
Using the Change of Base Formula to convert the logarithms in the given equation to base <math>2</math> yields | Using the Change of Base Formula to convert the logarithms in the given equation to base <math>2</math> yields |
Revision as of 14:20, 9 June 2020
Contents
Problem
The value of that satisfies
can be written as
, where
and
are relatively prime positive integers. Find
.
Solution
Let . Based on the equation, we get
and
. Expanding the second equation, we get
. Substituting the first equation in, we get
, so
. Taking the 100th root, we get
. Therefore,
, so
and the answer is
.
~rayfish
Easiest Solution
Recall the identity (which is easily proven using exponents or change of base)
Then this problem turns into
Divide
from both sides. And we are left with
.Solving this simple equation we get
~mlgjeffdoge21
Solution 2
Because we have that
or
Since
and
thus resulting in
or
We remove the base 3 logarithm and the power of 2 to yield
or
Our answer is
Solution 3 (Official MAA)
Using the Change of Base Formula to convert the logarithms in the given equation to base yields
Canceling the logarithm factors then yields
which has solution
The requested sum is
.
Video Solution
https://youtu.be/lPr4fYEoXi0 ~ CNCM
Video Solution 2
https://www.youtube.com/watch?v=x0QznvXcwHY?t=528
~IceMatrix
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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.