Question 247675
I don't know if either of the following are "rigorous". If not, repost your question.<ul><li>Using definitions and properties of inverses:<ol><li>As you say {{{e^x}}} and {{{ln(x)}}} are inverses of each other.</li><li>So {{{e^(ln(x))}}} is a composition of inverses.</li><li>The composition of <b>all</b> inverses results in the identity function: f(x) = x.</li></ol></li><li>Using Algebra:<ol><li>Find the natural logarithm of each side:
{{{ln(e^(ln(x))) = ln(x)}}}</li><li>Use the property of logarithms, {{{log(a, (p^q)) = q*log(a, (p))}}} to move the exponent of the argument in front of the logarithm:
{{{ln(x)*ln(e) = ln(x)}}}</li><li>By definition, ln(e) = 1:
{{{ln(x)*1 = ln(x)}}}</li><li>By the Identity Property of Multiplication the left side simplifies to:
{{{ln(x) = ln(x)}}} which is true for all positive values of x.</li></ol></li></ul>