AIME 2020 I · 第 2 题

Solution 1

Since these form a geometric series, $\frac{\log_2{x}}{\log_4{x}}$ is the common ratio. Rewriting this, we get $\frac{\log_x{4}}{\log_x{2}} = \log_2{4} = 2$ by base change formula. Therefore, the common ratio is 2. Now $\frac{\log_4{x}}{\log_8{2x}} = 2 \implies \log_4{x} = 2\log_8{2} + 2\log_8{x} \implies \frac{1}{2}\log_2{x} = \frac{2}{3} + \frac{2}{3}\log_2{x}$

$\implies -\frac{1}{6}\log_2{x} = \frac{2}{3} \implies \log_2{x} = -4 \implies x = \frac{1}{16}$. Therefore, $1 + 16 = \boxed{017}$.

~ JHawk0224

Solution 2

If we set $x=2^y$, we can obtain three terms of a geometric sequence through logarithm properties. The three terms are

$$\frac{y+1}{3}, \frac{y}{2}, y.$$ In a three-term geometric sequence, the middle term squared is equal to the product of the other two terms, so we obtain the following:

$$\frac{y^2+y}{3} = \frac{y^2}{4},$$ which can be solved to reveal $y = -4$. Therefore, $x = 2^{-4} = \frac{1}{16}$, so our answer is $\boxed{017}$.

-molocyxu

Solution 3

Let $r$ be the common ratio. We have

$$r = \frac{\log_4{(x)}}{\log_8{(2x)}} = \frac{\log_2{(x)}}{\log_4{(x)}}$$ Hence we obtain

$$(\log_4{(x)})(\log_4{(x)}) = (\log_8{(2x)})(\log_2{(x)})$$ Ideally we change everything to base $64$ and we can get:

$$(\log_{64}{(x^3)})(\log_{64}{(x^3)}) = (\log_{64}{(x^6)})(\log_{64}{(4x^2)})$$ Now divide to get:

$$\frac{\log_{64}{(x^3)}}{\log_{64}{(4x^2)}} = \frac{\log_{64}{(x^6)}}{\log_{64}{(x^3)}}$$ By change-of-base we obtain:

$$\log_{(4x^2)}{(x^3)} = \log_{(x^3)}{(x^6)} = 2$$ Hence $(4x^2)^2 = x^3 \rightarrow 16x^4 = x^3 \rightarrow x = \frac{1}{16}$ and we have $1+16 = \boxed{017}$ as desired.

~skyscraper

Solution 4 (Exponents > Logarithms)

Let $r$ be the common ratio, and let $a$ be the starting term ($a=\log_{8}{(2x)}$). We then have:

$$\log_{8}{(2x)}=a, \log_{4}{(x)}=ar, \log_{2}{(x)}=ar^2$$ Rearranging these equations gives:

$$8^a=2x, 4^{ar}=x, 2^{ar^2}=x$$ Deal with the last two equations first: Setting them equal gives:

$$4^{ar}=2^{ar^2} \implies 2^{2ar}=2^{ar^2} \implies 2ar=ar^2 \implies r=2$$ Using this value of $r$, substitute into the first and second equations (or the first and third, it doesn't really matter) to get:

$$8^a=2x, 4^{2a}=x$$ Changing these to a common base gives:

$$2^{3a}=2x, 2^{4a}=x$$ Dividing the first equation by 2 on both sides yields:

$$2^{3a-1}=x$$ Setting these equations equal to each other and removing the exponent again gives:

$$3a-1=4a \implies a=-1$$ Substituting this back into the first equation gives:

$$8^{-1}=2x \implies 2x=\frac{1}{8} \implies x=\frac{1}{16}$$ Therefore, $m+n=1+16=\boxed{017}$

~IAmTheHazard

Solution 5

We can relate the logarithms as follows:

$$\frac{\log_4{x}}{\log_8{(2x)}}=\frac{\log_2{x}}{\log_4{x}}$$ $$\log_8{(2x)}\log_2{x}=\log_4{x}\log_4{x}$$ Now we can convert all logarithm bases to $2$ using the identity $\log_a{b}=\log_{a^c}{b^c}$:

$$\log_2{\sqrt[3]{2x}}\log_2{x}=\log_2{\sqrt{x}}\log_2{\sqrt{x}}$$ We can solve for $x$ as follows:

$$\frac{1}{3}\log_2{(2x)}\log_2{x}=\frac{1}{4}\log_2{x}\log_2{x}$$ $$\frac{1}{3}\log_2{(2x)}=\frac{1}{4}\log_2{x}$$ $$\frac{1}{3}\log_2{2}+\frac{1}{3}\log_2{x}=\frac{1}{4}\log_2{x}$$ We get $x=\frac{1}{16}$. Verifying that the common ratio is positive, we find the answer of $\boxed{017}$.

~QIDb602

Solution 6

If the numbers are in a geometric sequence, the middle term must be the geometric mean of the surrounding terms. We can rewrite the first two logarithmic expressions as $\frac{1+\log_2{x}}{3}$ and $\frac{1}{2}\log_2{x}$, respectively. Therefore:

$$\frac{1}{2}\log_2{x}=\sqrt{\left(\frac{1+\log_2{x}}{3}\right)\left(\log_2{x}\right)}$$ Let $n=\log_2{x}$. We can rewrite the expression as:

$$\frac{n}{2}=\sqrt{\frac{n(n+1)}{3}}$$ $$\frac{n^2}{4}=\frac{n(n+1)}{3}$$ $$4n(n+1)=3n^2$$ $$4n^2+4n=3n^2$$ $$n^2+4n=0$$ $$n(n+4)=0$$ $$n=0 \text{ and } -4$$ Zero does not work in this case, so we consider $n=-4$: $\log_2{x}=-4 \rightarrow x=\frac{1}{16}$. Therefore, $1+16=\boxed{017}$.

~Bowser498

Solution 7

Again, by the Change of Base Formula, obtain that the common ratio is 2. If we let $y$ be the exponent of $\log_8 (2x)$, then we have $8^y=2x;\:4^{2y}=x;\:2^{4y}=x.$ Wee can then divide the first equation by two to have the right side be $x$. Also, $2^{4y}=\left(2^{4}\right)^y=16^y$. Setting this equal to $\frac{8^y}{2}$, we can divide the two equations to get $2^y=\frac12$. Therefore, $y=-1$. After that, we can raise $16$ to the $-1$th power to get $x=16^{-1}\Rightarrow x=\frac1{16}$. We then get our sum of $1+16=\boxed{\textbf{017}}$.

~SweetMango77

Solution 8 (Official MAA)

By the Change of Base Formula the common ratio of the progression is

$$\frac{\log_2 x}{\log_4 x} = \frac{\hphantom{m}\log_2x\hphantom{m}}{\frac{\log_2x}{\log_24}} = 2.$$ Hence $x$ must satisfy

$$2=\frac{\log_4 x}{\log_8 (2x)}= \frac{\log_2 x}{\log_2 4} \div \frac{\log_2(2x)}{\log_28} = \frac 32\cdot \frac{\log_2x}{1+\log_2x}.$$ This is equivalent to $4 + 4\log_2x = 3\log_2x$. Hence $\log_2x = -4$ and $x = \frac{1}{16}$. The requested sum is $1+16 = 17$.

Solution 9

Let the series be $a,ar,ar^2$. So, converting these logarithms into exponents we have $8^a = 2x$, $4^{ar} = x$, and $2^{ar^2} = x$. So, setting the 2nd and 3rd equations equal, we have:

$$4^{ar} = 2^{ar^2} \Longrightarrow 2^{2ar} = 2^{ar^2} \Longrightarrow 2ar = ar^2 \Longrightarrow r = 2$$ . Plugging that into our 3rd equation, we have $2^{4a} = x$. We recall that $8^a = 2x \Longrightarrow 2^{3a} = 2x$. So, since $2^{4a} = x$, $2x = 2^{4a+1}$. Now, setting $2^{3a}$, and $2^{4a+1}$ equal, we get $a = -1$, so plugging into our first equation, we have

$$\dfrac{1}{8} = 2x \Longrightarrow x = \dfrac{1}{16} \Longrightarrow \boxed{017}$$ -jb2015007

Video Solutions

https://youtu.be/nPL7nUXnRbo (Unavailable)

Aarav Navani: https://youtu.be/4FvYVfhhTaQ

Mathematica De Estremo: https://youtu.be/FgrIgCyGVUI

https://youtu.be/mgRNqSDCvgM?t=281s (Unavailable)