Intutively, why does x 1 ln ⁡ x = e? x &gt; 0, we have this identity: x 1 ln ⁡ x = e .You can see this by using the fact that x = e ln ⁡ x I'm wondering if there's a good intuitive explanation for this one, given that x 1 k is the operation that inverts raising x to the kth power and ln ⁡ x is the inverse of the exponential function. Is there some compelling intuitive or geometric argument that makes this identity more obvious than algebraic rearrangement?

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Intutively, why does       x                  1                  ln          ⁡          x                      =  e?  x  &amp;gt;  0, we have this identity:      x                  1                  ln          ⁡          x                      =  e  .You can see this by using the fact that   x  =      e          ln      ⁡      x      I&#039;m wondering if there&#039;s a good intuitive explanation for this one, given that       x                  1        k             is the operation that inverts raising   x to the   kth power and   ln  ⁡  x is the inverse of the exponential function. Is there some compelling intuitive or geometric argument that makes this identity more obvious than algebraic rearrangement?

Eva Cochran · Accepted Answer

The expression   log  ⁡  x describes a number that is an exponent: specifically, it is the exponent when, to which   e is raised, yields   x. When we speak of   1      /    log  ⁡  x, this is in some sense an inverse exponent: for example, if I write      a    b    =  c  ,then      c          1              /            b        =  a  .So   1      /    b is the exponent to which   c is raised to &quot;recover&quot;   a. Similarly, then, as   log  ⁡  x is the exponent to which   e is raised to yield   x, we must then have   1      /    log  ⁡  x as the exponent to which   x must be raised to yield   e.Of course, all of the above is nothing more than a more prosaic restatement of the algebraic rules that you already wrote in your question.

inurbandojoa · Accepted Answer

x                            (                      1                          ln              ⁡              x                                )                      =  e    ⟺              (              x                                            (                              1                                  ln                  ⁡                  x                                            )                                          )              ln      ⁡      x        =      e          ln      ⁡      x          ⟺        x    1    =      x          ln      ⁡      e        =      x    1    ≡      T

Intutively, why does x^(1/(ln x)) = e ? For any x>0, we have this identity: x^(1/(ln x)) = e You can see this by using the fact that x=e^(ln x).

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