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

Revision as of 11:14, 13 August 2021

Problem

The solutions to the equation $\log_{3x} 4 = \log_{2x} 8$ , where $x$ is a positive real number other than $\tfrac{1}{3}$ or $\tfrac{1}{2}$ , can be written as $\tfrac {p}{q}$ 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 $$ (Error compiling LaTeX. Unknown error_msg)\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

If you multiply both sides by $\log_2 (3x)$

then it should come out to $\log_2 (3x)$ * $\log_{3x} (4)$ = $\log_2 {3x}$ * $\log_{2x} (8)$

that then becomes $\log_2 (4)$ * $\log_{3x} (3x)$ = $\log_2 (8)$ * $\log_{2x} (3x)$

which simplifies to $2*1 = 3\log_{2x} (3x)$

so now $\frac{2}{3}$ = $\log_{2x} (3x)$ putting in exponent form gets

$(2x)^2$ = $(3x)^3$

so $4x^2$ = $27x^3$

dividing yields $x = 4/27$ and

$4+27 =$ $\boxed{\textbf{(D)}31}$

- Pikachu13307

Solution 3

We can convert both $4$ and $8$ into $2^2$ and $2^3$ , respectively, giving:

$2\log_{3x} (2) = 3\log_{2x} (2)$

Converting the bases of the right side, we get $\log_{2x} 2 = \frac{\ln 2}{\ln (2x)}$

$\frac{2}{3}*\log_{3x} (2) = \frac{\ln 2}{\ln (2x)}$

$2^\frac{2}{3} = (3x)^\frac{\ln 2}{\ln (2x)}$

$\frac{2}{3} * \ln 2 = \frac{\ln 2}{\ln (2x)} * \ln (3x)$

Dividing both sides by $\ln 2$ , we get

$\frac{2}{3} = \frac{\ln (3x)}{\ln (2x)}$

Which simplifies to

$2\ln (2x) = 3\ln (3x)$

Log expansion allows us to see that

$2\ln 2 + 2\ln (x) = 3\ln 3 + 3\ln (x)$ , which then simplifies to

$\ln (x) = 2\ln 2 - 3\ln 3$

Thus,

$x = e^{2\ln 2 - 3\ln 3} = \frac{e^{2\ln 2}}{e^{3\ln 3}}$

And

$x = \frac{2^2}{3^3} = \frac{4}{27} = \boxed{\textbf{(D)}31}$

-lepetitmoulin

Solution 4

$\log_{3x} 4=\log_{2x} 8$ is the same as $2\log_{3x} 2=3\log_{2x} 2$

Using Reciprocal law, we get $\log_{(3x)^\frac{1}{2}} 2=\log_{(2x)^\frac{1}{3}} 2$

$\Rightarrow (3x)^\frac{1}{2}=(2x)^\frac{1}{3}$ $\Rightarrow 27x^3=4x^2$ $\Rightarrow \frac{x^3}{x^2}=\frac{4}{27}=x$

$\therefore \frac{p}{q}=\frac{4}{27}$ $\Rightarrow p+q=4+27=$ $\boxed{\textbf{(D) } 31}$

~OlutosinNGA

Solution 5

$\log_{3x} 4=\log_{2x} 8\implies 2\log_{3x} 2=3\log_{2x} 2 \implies \frac{2}{3}=\frac{\log_{2x}2}{\log_{2x}3}$ . We know that $\log_a{b}=\frac{\log_{c}b}{\log_{c}a}=\frac{\frac{1}{\log_b{c}}}{\frac{1}{\log_a{c}}}=\frac{\log_a{c}}{\log_b{c}}$ . Thus $\frac{2}{3}=\frac{\log_{2x}2}{\log_{2x}3}\implies \frac{2}{3}=\log_{2x}{3x}\implies (2x)^{\frac{2}{3}}=3x\implies 2^{\frac{2}{3}}x^{\frac{2}{3}}=3x\implies 2^{\frac{2}{3}}=3x^{\frac{1}{3}}\implies x^{\frac{1}{3}}=\frac{2^{\frac{2}{3}}}{3}\implies x=\frac{2^2}{3^3}=\frac{4}{27}$ . $4$ and $27$ are indeed relatively prime thus our final answer is $4+27=31 \text{which is }\boxed{\textbf{(D)}}$

-vsamc

@@ 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
+<math></math>\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*}
+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==

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