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

Revision as of 14:42, 23 March 2018

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

Base switch to log 2 and you have $\frac{\log_2 4}{\log_2 3x} = \frac{\log_2 8}{\log_2 2x}$ .

$\frac{2}{\log_2 3x} = \frac{3}{\log_2 2x}$

$2*\log_2 2x = 3*\log_2 3x$

Then $\log_2 (2x)^2 = \log_2 (3x)^3$ . so $4x^2=27x^3$ and we have $x=\frac{4}{27}$ leading to $\boxed{ (D) 31}$ (jeremylu)

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

dividing yields x = 4/27 and

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

- 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

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

Eventually, we find that

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

And

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

-lepetitmoulin

@@ Line 39: / Line 39: @@
 +27 = <math>\boxed{ (D) 31}</math>
   - Pikachu13307
+==Solution 3==
+We can convert both <math>4</math> and <math>8</math> into <math>2^2</math> and <math>2^3</math>, respectively, giving:
+<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>
+Eventually, we find that
+<math>x = e^{2\ln 2 - 3\ln 3} = \frac{e^{2\ln 2}}{e^{3\ln 3}}</math>
+And
+<math>x = \frac{2^2}{3^3} = \frac{4}{27} = \boxed{ (D) 31}</math>
+-lepetitmoulin
 ==See Also==
 {{AMC12 box|year=2018|ab=A|num-b=13|num-a=15}}
 {{MAA Notice}}

2018 AMC 12A (Problems • Answer Key • Resources)
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