Logarithmic system of equations <mtable columnalign="right left right left right left right lef

kolutastmr 2022-07-02 Answered
Logarithmic system of equations
ln ( x ) = 3 ln ( y )   3 x = 27 y
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Answers (2)

fugprurgeil
Answered 2022-07-03 Author has 12 answers
Notice
ln x = 2 ln y ln x = ln y 2 x = y 2
and
3 x = 27 y 3 x = 3 3 y 3 x 3 = y 3 y 2 3 = y ln 3 ( y 2 3 ) = y y 2 3 y ln 3 = 0 y 2 1 ln 3 y 3 = 0
This is a quadratic equation, which you can easily solve.

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Palmosigx
Answered 2022-07-04 Author has 4 answers
The first equation says that x , y > 0 and then it's equivalent to x = e ln x = e 3 ln y = e ln y 3 = y 3 .
The second equation is equivalent to x = 3 y by taking a logarithm, so we have
3 y = y 3 y ( y 3 ) ( y + 3 ) = 0
and the only solution is y = 3 , which gives x = 3 3 .

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