Difference between revisions of "2018 AMC 12A Problems/Problem 14"

Latest revision as of 11:37, 23 July 2024

Problem

The solution to the equation $\log_{3x} 4 = \log_{2x} 8$ , where $x$ is a positive real number other than $\frac{1}{3}$ or $\frac{1}{2}$ , can be written as $\frac {p}{q}$ where $p$ and $q$ are relatively prime positive integers. What is $p + q$ ?

$\textbf{(A) } 5 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 17 \qquad \textbf{(D) } 31 \qquad \textbf{(E) } 35$

Solution 1

We apply the Change of Base Formula, then rearrange: $\begin{align*} \frac{\log_2{4}}{\log_2{(3x)}}&=\frac{\log_2{8}}{\log_2{(2x)}} \\ \frac{2}{\log_2{(3x)}}&=\frac{3}{\log_2{(2x)}} \\ 3\log_2{(3x)}&=2\log_2{(2x)}. \\ \end{align*}$ By the logarithmic identity $n\log_b{a}=\log_b{\left(a^n\right)},$ it follows that $\begin{align*} \log_2{\left[(3x)^3\right]}&=\log_2{\left[(2x)^2\right]} \\ (3x)^3&=(2x)^2 \\ 27x^3&=4x^2 \\ x&=\frac{4}{27}, \end{align*}$ from which the answer is $4+27=\boxed{\textbf{(D) } 31}.$

~jeremylu (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

Solution 2

We will apply the following logarithmic identity: $\log_{p^n}{\left(q^n\right)}=\log_{p}{q},$ which can be proven by the Change of Base Formula: $\log_{p^n}{\left(q^n\right)}=\frac{\log_{p}{\left(q^n\right)}}{\log_{p}{\left(p^n\right)}}=\frac{n\log_{p}{q}}{n}=\log_{p}{q}.$ We rewrite the original equation as $\log_{(3x)^3} 64 = \log_{(2x)^2} 64,$ from which $\begin{align*} (3x)^3&=(2x)^2 \\ 27x^3&=4x^2 \\ x&=\frac{4}{27}. \end{align*}$ Therefore, the answer is $4+27=\boxed{\textbf{(D) } 31}.$

~MRENTHUSIASM

Solution 3

By the logarithmic identity $n\log_b{a}=\log_b{\left(a^n\right)},$ the original equation becomes $2\log_{3x} 2 = 3\log_{2x} 2.$ By the logarithmic identity $\log_b{a}\cdot\log_a{b}=1,$ we multiply both sides by $\log_2{(2x)},$ then apply the Change of Base Formula to the left side: $\begin{align*} 2\left[\log_{3x}2\right]\left[\log_2{(2x)}\right] &= 3 \\ 2\left[\frac{\log_2 2}{\log_2{(3x)}}\right]\left[\frac{\log_2{(2x)}}{\log_2 2}\right] &= 3 \\ 2\left[\frac{\log_2{(2x)}}{\log_2{(3x)}}\right] &=3 \\ 2\left[\log_{3x}{(2x)}\right] &= 3 \\ \log_{3x}{\left[(2x)^2\right]} &= 3 \\ (3x)^3&=(2x)^2 \\ 27x^3&=4x^2 \\ x&=\frac{4}{27}. \end{align*}$ Therefore, the answer is $4+27=\boxed{\textbf{(D) } 31}.$

~Pikachu13307 (Fundamental Logic)

~MRENTHUSIASM (Reconstruction)

Solution 4

We can convert both $4$ and $8$ into $2^2$ and $2^3,$ respectively: $2\log_{3x} (2) = 3\log_{2x} (2).$ Converting the bases of the right side, we get $\begin{align*} \log_{2x} 2 &= \frac{\ln 2}{\ln (2x)} \\ \frac{2}{3}\cdot\log_{3x} (2) &= \frac{\ln 2}{\ln (2x)} \\ 2^\frac{2}{3} &= (3x)^\frac{\ln 2}{\ln (2x)} \\ \frac{2}{3} \cdot \ln 2 &= \frac{\ln 2}{\ln (2x)} \cdot \ln (3x). \end{align*}$ Dividing both sides by $\ln 2,$ we get $\frac{2}{3} = \frac{\ln (3x)}{\ln (2x)},$ from which $2\ln (2x) = 3\ln (3x).$ Expanding this equation gives $\begin{align*} 2\ln 2 + 2\ln (x) &= 3\ln 3 + 3\ln (x) \\ \ln (x) &= 2\ln 2 - 3\ln 3. \end{align*}$ Thus, we have $x = e^{2\ln 2 - 3\ln 3} = \frac{e^{2\ln 2}}{e^{3\ln 3}} = \frac{2^2}{3^3} = \frac{4}{27},$ from which the answer is $4+27=\boxed{\textbf{(D) } 31}.$

~lepetitmoulin (Solution)

~MRENTHUSIASM (Reformatting)

Solution 5 (Exponential Form)

Let $y=\log_{3x} 4 = \log_{2x} 8.$ We convert the equations with $y$ to the exponential form: $\begin{align*} (3x)^y&=4, \\ (2x)^y&=8. \end{align*}$ Cubing the first equation and squaring the second equation, we have $\begin{align*} (3x)^{3y}&=64, \\ (2x)^{2y}&=64. \end{align*}$ Applying the Transitive Property, we get $\begin{align*} (3x)^{3y}&=(2x)^{2y} \\ (3x)^3&=(2x)^2 \\ 27x^3&=4x^2 \\ x&=\frac{4}{27}, \end{align*}$ from which the answer is $4+27=\boxed{\textbf{(D) } 31}.$

~MRENTHUSIASM

@@ Line 1: / Line 1: @@
 ==Problem==
-The solutions to the equation <math>\log_{3x} 4 = \log_{2x} 8</math>, where <math>x</math> is a positive real number other than <math>\tfrac{1}{3}</math> or <math>\tfrac{1}{2}</math>, can be written as <math>\tfrac {p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p + q</math>?
+The solution to the equation <math>\log_{3x} 4 = \log_{2x} 8</math>, where <math>x</math> is a positive real number other than <math>\frac{1}{3}</math> or <math>\frac{1}{2}</math>, can be written as <math>\frac {p}{q}</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p + q</math>?
 <math>\textbf{(A) } 5   \qquad
@@ Line 10: / Line 10: @@
 ==Solution 1==
+We apply the Change of Base Formula, then rearrange:
+<cmath>\begin{align*}
+\frac{\log_2{4}}{\log_2{(3x)}}&=\frac{\log_2{8}}{\log_2{(2x)}} \\
+\frac{2}{\log_2{(3x)}}&=\frac{3}{\log_2{(2x)}} \\
+\log_2{(3x)}&=2\log_2{(2x)}. \\
+\end{align*}</cmath>
+By the logarithmic identity <math>n\log_b{a}=\log_b{\left(a^n\right)},</math> it follows that
+<cmath>\begin{align*}
+\log_2{\left[(3x)^3\right]}&=\log_2{\left[(2x)^2\right]} \\
+(3x)^3&=(2x)^2 \\
+x^3&=4x^2 \\
+x&=\frac{4}{27},
+\end{align*}</cmath>
+from which the answer is <math>4+27=\boxed{\textbf{(D) } 31}.</math>
+~jeremylu (Fundamental Logic)
-Base switch to log 2 and you have <math>\frac{\log_2 4}{\log_2 3x} = \frac{\log_2 8}{\log_2 2x}</math> .
+~MRENTHUSIASM (Reconstruction)
-<math>\frac{2}{\log_2 3x} = \frac{3}{\log_2 2x}</math>
-<math>2*\log_2 2x = 3*\log_2 3x</math>
-Then <math>\log_2 (2x)^2 = \log_2 (3x)^3</math>. so <math>4x^2=27x^3</math> and we have <math>x=\frac{4}{27}</math> leading to <math>\boxed{\textbf{(D)}31}</math> (jeremylu)
 ==Solution 2==
-If you multiply both sides by <math>\log_2 (3x)</math>
+We will apply the following logarithmic identity:
+<cmath>\log_{p^n}{\left(q^n\right)}=\log_{p}{q},</cmath>
-then it should come out to <math>\log_2 (3x)</math> * <math>\log_{3x} (4)</math> = <math>\log_2 {3x}</math> * <math>\log_{2x} (8)</math>
+which can be proven by the Change of Base Formula: <cmath>\log_{p^n}{\left(q^n\right)}=\frac{\log_{p}{\left(q^n\right)}}{\log_{p}{\left(p^n\right)}}=\frac{n\log_{p}{q}}{n}=\log_{p}{q}.</cmath>
+We rewrite the original equation as <math>\log_{(3x)^3} 64 = \log_{(2x)^2} 64,</math> from which
+<cmath>\begin{align*}
+(3x)^3&=(2x)^2 \\
+x^3&=4x^2 \\
+x&=\frac{4}{27}.
+\end{align*}</cmath>
+Therefore, the answer is <math>4+27=\boxed{\textbf{(D) } 31}.</math>
-that then becomes <math>\log_2 (4)</math> * <math>\log_{3x} (3x)</math> = <math>\log_2 (8)</math> * <math>\log_{2x} (3x)</math>
+~MRENTHUSIASM
-which simplifies to <math>2*1 = 3\log_{2x} (3x)</math>
-so now <math>\frac{2}{3}</math> = <math>\log_{2x} (3x)</math> putting in exponent form gets
-<math>(2x)^2</math> = <math>(3x)^3</math>
-so <math>4x^2</math> = <math>27x^3</math>
-dividing yields <math>x = 4/27</math> and
-<math>4+27 =</math> <math>\boxed{\textbf{(D)}31}</math>
- - Pikachu13307
 ==Solution 3==
+By the logarithmic identity <math>n\log_b{a}=\log_b{\left(a^n\right)},</math> the original equation becomes <cmath>2\log_{3x} 2 = 3\log_{2x} 2.</cmath>
+By the logarithmic identity <math>\log_b{a}\cdot\log_a{b}=1,</math> we multiply both sides by <math>\log_2{(2x)},</math> then apply the Change of Base Formula to the left side:
+<cmath>\begin{align*}
+\left[\log_{3x}2\right]\left[\log_2{(2x)}\right] &= 3 \\
+\left[\frac{\log_2 2}{\log_2{(3x)}}\right]\left[\frac{\log_2{(2x)}}{\log_2 2}\right] &= 3 \\
+\left[\frac{\log_2{(2x)}}{\log_2{(3x)}}\right] &=3 \\
+\left[\log_{3x}{(2x)}\right] &= 3 \\
+\log_{3x}{\left[(2x)^2\right]} &= 3 \\
+(3x)^3&=(2x)^2 \\
+x^3&=4x^2 \\
+x&=\frac{4}{27}.
+\end{align*}</cmath>
+Therefore, the answer is <math>4+27=\boxed{\textbf{(D) } 31}.</math>
-We can convert both <math>4</math> and <math>8</math> into <math>2^2</math> and <math>2^3</math>, respectively, giving:
+~Pikachu13307 (Fundamental Logic)
-<math>2\log_{3x} (2) = 3\log_{2x} (2)</math>
-Converting the bases of the right side, we get <math>\log_{2x} 2 = \frac{\ln 2}{\ln (2x)}</math>
-<math>\frac{2}{3}*\log_{3x} (2) = \frac{\ln 2}{\ln (2x)}</math>
-<math>2^\frac{2}{3} = (3x)^\frac{\ln 2}{\ln (2x)}</math>
-<math>\frac{2}{3} * \ln 2 = \frac{\ln 2}{\ln (2x)} * \ln (3x)</math>
-Dividing both sides by <math>\ln 2</math>, we get
-<math>\frac{2}{3} = \frac{\ln (3x)}{\ln (2x)}</math>
-Which simplifies to
-<math>2\ln (2x) = 3\ln (3x)</math>
-Log expansion allows us to see that
-<math>2\ln 2 + 2\ln (x) = 3\ln 3 + 3\ln (x)</math>, which then simplifies to
-<math>\ln (x) = 2\ln 2 - 3\ln 3</math>
+~MRENTHUSIASM (Reconstruction)
-Thus,
+==Solution 4==
+We can convert both <math>4</math> and <math>8</math> into <math>2^2</math> and <math>2^3,</math> respectively: <cmath>2\log_{3x} (2) = 3\log_{2x} (2).</cmath>
+Converting the bases of the right side, we get
+<cmath>\begin{align*}
+\log_{2x} 2 &= \frac{\ln 2}{\ln (2x)} \\
+\frac{2}{3}\cdot\log_{3x} (2) &= \frac{\ln 2}{\ln (2x)} \\
+^\frac{2}{3} &= (3x)^\frac{\ln 2}{\ln (2x)} \\
+\frac{2}{3} \cdot \ln 2 &= \frac{\ln 2}{\ln (2x)} \cdot \ln (3x).
+\end{align*}</cmath>
+Dividing both sides by <math>\ln 2,</math> we get <math>\frac{2}{3} = \frac{\ln (3x)}{\ln (2x)},</math> from which <cmath>2\ln (2x) = 3\ln (3x).</cmath>
+Expanding this equation gives
+<cmath>\begin{align*}
+\ln 2 + 2\ln (x) &= 3\ln 3 + 3\ln (x) \\
+\ln (x) &= 2\ln 2 - 3\ln 3.
+\end{align*}</cmath>
+Thus, we have <cmath>x = e^{2\ln 2 - 3\ln 3} = \frac{e^{2\ln 2}}{e^{3\ln 3}} = \frac{2^2}{3^3} = \frac{4}{27},</cmath>
+from which the answer is <math>4+27=\boxed{\textbf{(D) } 31}.</math>
-<math>x = e^{2\ln 2 - 3\ln 3} = \frac{e^{2\ln 2}}{e^{3\ln 3}}</math>
+~lepetitmoulin (Solution)
-And
+~MRENTHUSIASM (Reformatting)
-<math>x = \frac{2^2}{3^3} = \frac{4}{27} = \boxed{\textbf{(D)}31}</math>
+==Solution 5 (Exponential Form)==
+Let <math>y=\log_{3x} 4 = \log_{2x} 8.</math> We convert the equations with <math>y</math> to the exponential form:
+<cmath>\begin{align*}
+(3x)^y&=4, \\
+(2x)^y&=8.
+\end{align*}</cmath>
+Cubing the first equation and squaring the second equation, we have
+<cmath>\begin{align*}
+(3x)^{3y}&=64, \\
+(2x)^{2y}&=64.
+\end{align*}</cmath>
+Applying the Transitive Property, we get
+<cmath>\begin{align*}
+(3x)^{3y}&=(2x)^{2y} \\
+(3x)^3&=(2x)^2 \\
+x^3&=4x^2 \\
+x&=\frac{4}{27},
+\end{align*}</cmath>
+from which the answer is <math>4+27=\boxed{\textbf{(D) } 31}.</math>
--lepetitmoulin
+~MRENTHUSIASM
 ==See Also==
 {{AMC12 box|year=2018|ab=A|num-b=13|num-a=15}}
 {{MAA Notice}}

2018 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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