How do we solve a ≤ b r − r for r?

Question

How do we solve   a  ≤      b          r        −  r for r?

Krish Finley · Accepted Answer

Step 1The case of equality can be solved in terms of the Lambert W function.It can be used to solve equations of the form      p          a      x      +      b        =  c  x  +  d(quoted from the wiki page).      b          r        −  r is &#039;mostly&#039; monotonic, so I suppose solving the equality will be enough to find the solutions to your inequality.

Denisse Valdez · Accepted Answer

Step 1You can also get a quick-and-dirty approximation for the equality case with the following approach, which should hold in most situations when   a  ,  b  &amp;gt;  1 .Rearrange the problem and take logarithms to rewrite it as  ln  ⁡      (    1    +          r      a        )    =  r  ln  ⁡  b  −  ln  ⁡  a  .Now, if       r    a   is less than 1 (and it should be in most cases where   a  ,  b  &amp;gt;  1 ),  ln  ⁡      (    1    +          r      a        )    ≈      r    a    , ,which gives you  r  ≈            −      ln      ⁡      a              1              /            a      −      ln      ⁡      b        . .For your specific problem, this method yields   r  ≈  3.78 , whereas the actual answer is closer to 3.66.You can improve this by using the quadratic approximation  ln  ⁡      (    1    +          r      a        )    ≈      r    a    −            r      2              2              a        2              , ,which requires solving a quadratic equation.(These approximations come from the Taylor expansion of   ln  ⁡  (  1  +  x  ) .)

How do we solve a ≤<!-- ≤

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