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Find the constant \(k\) so that \(\log_{y^5}(x^3) = k \cdot\log_y(x)\) for all positive real numbers \(x\) and \(y\) with \(y \neq 1.\) 

 Apr 10, 2019
 #1
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log y^5 (x^3)  =  k log y (x)

 

Using the change-of base theorem

 

log x^3              log x^k

______   =       ________

log y^5               log y

 

 

3 log x                 k log x

______ =           _______

5 log y                  log y

 

3 log x log y  =  5k log x log y

 

3 = 5k

 

k = 3/5

 

cool cool cool

 Apr 10, 2019

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