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Suppose that x and y are positive numbers such that \(\log_x y - \log_y x = 3\)

 

Find \((\log_x y)^2 + (\log_y x)^2\)

 May 12, 2020
 #1
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Square  both   sides

 

[ log x y]^2  - 2 [ log x y ][log y x ]  + [ log y x ]^2  =  9

 

Using the change of  base theorem we have that

 

log x y  *  log x  =

 

[log y / log x] * [ log x / log y]   =  1

 

Sowe have  that

 

[ log x y ]^2   - 2 (1)  + [ log y x]^2  =  9            add 2 to both sides

 

[ log x y ]^2   +  [ log y x ]^2  =   11

 

 

cool cool cool

 May 12, 2020

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