Question

Sean

x,

y,

y

z

mayores que

1

, y sea

w

un número positivo tal que

\begin{aligned} \log_x w &= 24,\\ \log_y w &= 40,\\ \log_{xyz} w &= 12. \end{aligned}

Halla

\log_z w.

Let

x,

y,

and

z

all exceed

1

and let

w

be a positive number such that

\begin{aligned} \log_x w &= 24,\\ \log_y w &= 40,\\ \log_{xyz} w &= 12. \end{aligned}

Find

\log_z w.

Respuesta

Reescribe cada logaritmo dado con base

w

Rewrite each given logarithm with base

w

Expande

\log_w(xyz)

como una suma de tres logaritmos

Expand

\log_w(xyz)

as a sum of three logarithms

Al tomar los recíprocos de los logaritmos dados se obtiene

\begin{aligned} \log_w x&=\frac1{24},\\ \log_w y&=\frac1{40},\\ \log_w(xyz)&=\frac1{12}. \end{aligned}

Por lo tanto,

\log_w z=\frac1{12}-\frac1{24}-\frac1{40} =\frac1{60}.

Al tomar el recíproco se obtiene

\log_z w=60.

Taking reciprocals of the given logarithms gives

\begin{aligned} \log_w x&=\frac1{24},\\ \log_w y&=\frac1{40},\\ \log_w(xyz)&=\frac1{12}. \end{aligned}

Therefore

\log_w z=\frac1{12}-\frac1{24}-\frac1{40} =\frac1{60}.

Taking the reciprocal yields

\log_z w=60.