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Find x if \log_2 (\log_3 x) = 2 \log_2 (\log_{27} x).

 Dec 29, 2023
 #1
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\(\log_2 (\log_3 x) = 2 \log_2 (\log_{27} x) \)

 

 

We can write

 

log2 (log 3x) = log2 (log27x)^2

 

The log bases are the same, so

 

log3x = (log 27 x)^2

 

log3 x =  (log 27 x) (log 27 x)               {apply change-of-base theroem}

 

log x / log 3 =  ( log x / log 27) (log x / log 27)      {assume x not equal to 1} {divide out log x}

 

1/ log 3  =  (log x/ log 27) ( 1 /log 27)

 

log 27 / log 3  =  log x / log 27

 

(log 27)^2 / log 3 = log x

 

(log 3^3)^2   / log 3  = log x

 

(3 log 3)^2 / (log 3)  =  log x

 

9 (log 3)^2 / (log 3)  = log x

 

9 log 3  = log x

 

log 3^9  = log x

 

x = 3^9 =  19683

 

 

cool cool cool

 Dec 30, 2023

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