Question 1095567
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<pre>
1.  First step: introduce new variable u = {{{e^x}}}.

    Then your equation takes the form

    3u = 5 + {{{8/u}}}.    (1)



2.  Second step:  multiply by "u" both sides:

     3u^2 = 5u + 8,   or

     3u^2 - 5u - 8 = 0.



3.  You got quadratic equation. Its roots are

    {{{u[1,2]}}} = {{{(5 +- sqrt(5^2 + 4*3*8))/(2*3)}}} = {{{(5 +- 11)/6}}}.


    a)  {{{u[1]}}} = {{{(5 + 11)/6}}} = {{{16/6}}} = {{{8/3}}};

        ========>  {{{e^x}}} = {{{8/3}}}  ====>  x = {{{ln(8/3)}}}.


    b)  {{{u[2]}}} = {{{(5-11)/6}}}  = -1.

        ========>   {{{e^x}}} = -1   ====> There is NO solution for x.


<U>Answer</U>.  The given equation has a unique solution  x = {{{ln(8/3)}}}.
</pre>

Solved.



Introducing new variable is the standard method of solving such equations.