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Solve \(\log_{16} x + \log_4 x = 4.5\)

 May 31, 2020
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 Solve for x:
log(x)/log(4)+log(x)/log(16)==4.5


4.5==9/2:
log(x)/log(4)+log(x)/log(16)==9/2


Rewrite the left hand side by combining fractions. log(x)/log(4)+log(x)/log(16) == ((log(4)+log(16)) log(x))/(log(4) log(16)):
((log(4)+log(16)) log(x))/(log(4) log(16))==9/2


Divide both sides by log(4)+log(16)/(log(4) log(16)):
log(x)==(9 log(4) log(16))/(2 (log(4)+log(16)))


Cancel logarithms by taking exp of both sides:
x==E^((9 log(4) log(16))/(2 (log(4)+log(16))))

x = 64

 May 31, 2020

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