<p>Here is the original answer in full:</p>
<blockquote>
<p>Other answers address the question about the natural logarithm directly.</p>
<p>In more generality, why would you expect that for an arbitrary function <span class="math-container">$f,$</span> the mean value of <span class="math-container">$f(x)$</span> should be <span class="math-container">$f(0.5)$</span>?</p>
</blockquote>
<p>It doesn't answer the question.  Indeed, it's explicitly framed as another question! That's ideal as a Socratic <em>comment,</em> but it definitely should not stay up as an answer.</p>
<p>Perhaps that post should have been converted to a comment, another option available at the time.  I would be glad to do that retroactively.</p>
