Question

Si

\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_xx^2, \end{aligned}

entonces

y

es igual a:

If

\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_xx^2, \end{aligned}

then

y

equals:

\dfrac92

9

18

27

81

Usa

(\log_a b)(\log_b c)=\log_a c

dos veces

Use

(\log_a b)(\log_b c)=\log_a c

twice

Simplifica todo el lado izquierdo a

\log_3y

Simplify the entire left side to

\log_3y

Los logaritmos se cancelan telescópicamente:

\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_3y. \end{aligned}

Además,

\log_xx^2=2,

así que

\log_3y=2

y

y=3^2=9.

Por lo tanto, la respuesta correcta es B.

The logarithms telescope:

\begin{aligned} &(\log_3x)(\log_x2x)(\log_{2x}y)\\ &\qquad=\log_3y. \end{aligned}

Also

\log_xx^2=2,

so

\log_3y=2

and

y=3^2=9.

Therefore, the correct answer is B.