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\(\begin{cases} \begin{align} \log_mw & = 12 \\ \log_nw & = 16 \\ \log_pw & = 36 \\ \log_{mnpq}w & = 72 \end{align} \end{cases} \)

 

Find \(\log_q w\)

 Jun 18, 2020
 #1
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We can rewrite this as 

 

\(\begin{cases}\log_w m = \dfrac1{12}\\\log_w n = \dfrac1{16}\\\log_w p = \dfrac1{36}\\\log_{w} mnpq = \dfrac1{72}\end{cases}\)

 

By log properties, 

\(\log_w m + \log_w n + \log_w p + \log_w q = \dfrac1{72}\\ \dfrac1{12} + \dfrac1{16} + \dfrac1{36} + \dfrac1{\log_q w} = \dfrac1{72}\)

 

The rest is trivial.

 

Ans: \(\log_q w = -\dfrac{144}{23}\)

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 Jun 18, 2020

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