Difference between revisions of "2020 AIME II Problems/Problem 3"

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Let <math>\log _{2^x}3^{20}=\log _{2^{x+3}}3^{2020}=n</math>. Based on the equation, we get <math>(2^x)^n=3^{20}</math> and <math>(2^{x+3})^n=3^{2020}</math>. Expanding the second equation, we get <math>8^n\cdot2^{xn}=3^{2020}</math>. Substituting the first equation in, we get <math>8^n\cdot3^{20}=3^{2020}</math>, so <math>8^n=3^{2000}</math>. Taking the 100th root, we get <math>8^{\frac{n}{100}}=3^{20}</math>. Therefore, <math>(2^{\frac{3}{100}})^n=3^{20}</math>, so <math>n=\frac{3}{100}</math> and the answer is <math>\boxed{103}</math>.
 
Let <math>\log _{2^x}3^{20}=\log _{2^{x+3}}3^{2020}=n</math>. Based on the equation, we get <math>(2^x)^n=3^{20}</math> and <math>(2^{x+3})^n=3^{2020}</math>. Expanding the second equation, we get <math>8^n\cdot2^{xn}=3^{2020}</math>. Substituting the first equation in, we get <math>8^n\cdot3^{20}=3^{2020}</math>, so <math>8^n=3^{2000}</math>. Taking the 100th root, we get <math>8^{\frac{n}{100}}=3^{20}</math>. Therefore, <math>(2^{\frac{3}{100}})^n=3^{20}</math>, so <math>n=\frac{3}{100}</math> and the answer is <math>\boxed{103}</math>.
 
~rayfish
 
~rayfish
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==Easiest Solution==
 
==Easiest Solution==
 
Recall the identity <math>\log_{a^n} b^{m} = \frac{m}{n}\log_{a} b </math> (which is easily proven using exponents or change of base)
 
Recall the identity <math>\log_{a^n} b^{m} = \frac{m}{n}\log_{a} b </math> (which is easily proven using exponents or change of base)
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~mlgjeffdoge21
 
~mlgjeffdoge21
  
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==Solution 2==
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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>
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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 13:18, 9 June 2020

Problem

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Solution

Let $\log _{2^x}3^{20}=\log _{2^{x+3}}3^{2020}=n$. Based on the equation, we get $(2^x)^n=3^{20}$ and $(2^{x+3})^n=3^{2020}$. Expanding the second equation, we get $8^n\cdot2^{xn}=3^{2020}$. Substituting the first equation in, we get $8^n\cdot3^{20}=3^{2020}$, so $8^n=3^{2000}$. Taking the 100th root, we get $8^{\frac{n}{100}}=3^{20}$. Therefore, $(2^{\frac{3}{100}})^n=3^{20}$, so $n=\frac{3}{100}$ and the answer is $\boxed{103}$. ~rayfish

Easiest Solution

Recall the identity $\log_{a^n} b^{m} = \frac{m}{n}\log_{a} b$ (which is easily proven using exponents or change of base) Then this problem turns into \[\frac{20}{x}\log_{2} 3 = \frac{2020}{x+3}\log_{2} 3\] Divide $\log_{2} 3$ from both sides. And we are left with $\frac{20}{x}=\frac{2020}{x+3}$.Solving this simple equation we get \[x = \tfrac{3}{100} \Rightarrow \boxed{103}\] ~mlgjeffdoge21

Solution 2

Because $\log_a{b^c}=c\log_a{b},$ we have that $20\log_{2^x} 3 = 2020\log_{2^{x+3}} 3,$ or $\log_{2^x} 3 = 101\log_{2^{x+3}} 3.$ Since $\log_a{b}=\dfrac{1}{\log_b{a},$ (Error compiling LaTeX. ! File ended while scanning use of \@genfrac.) $\log_{2^x} 3=\dfrac{1}{\log_{3} 2^x},$ and $101\log_{2^{x+3}} 3=101\dfrac{1}{\log_{3}2^{x+3}},$ thus resulting in $\log_{3}2^{x+3}=101\log_{3} 2^x,$ or $\log_{3}2^{x+3}=\log_{3} 2^{101x}.$ We remove the base 3 logarithm and the power of 2 to yield $x+3=101x,$ or $x=\dfrac{3}{100}.$

Our answer is $\boxed{3+100=103}.$

Solution 3 (Official MAA)

Using the Change of Base Formula to convert the logarithms in the given equation to base $2$ yields \[\frac{\log_2 3^{20}}{\log_2 2^x} = \frac{\log_2 3^{2020}}{\log_2 2^{x+3}}, \text{~ and then ~} \frac{20\log_2 3}{x\cdot\log_2 2} = \frac{2020\log_2 3}{(x+3)\log_2 2}.\]Canceling the logarithm factors then yields\[\frac{20}x = \frac{2020}{x+3},\]which has solution $x = \frac3{100}.$ The requested sum is $3 + 100 = 103$.

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 (ProblemsAnswer KeyResources)
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

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