Logarithmic system of equations ln ⁡ ( x ) = 3 ln ⁡ ( y ) 3 x = 27 y

Question

Logarithmic system of equations                    ln        ⁡        (        x        )                            =        3        ln        ⁡        (        y        )                                               3          x                                    =                  27          y

fugprurgeil · Accepted Answer

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  =  0This is a quadratic equation, which you can easily solve.

Palmosigx · Accepted Answer

The first equation says that   x  ,  y  &amp;gt;  0 and then it&#039;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    )  =  0and the only solution is   y  =      3  , which gives   x  =  3      3  .

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

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