document.write( "Question 247675: prove: e^(ln(x)) = x
\n" ); document.write( "I know that e and ln are inverses. I'd like a rigorous proof though. thanks \n" ); document.write( "
Algebra.Com's Answer #180604 by jsmallt9(3758)\"\" \"About 
You can put this solution on YOUR website!
I don't know if either of the following are \"rigorous\". If not, repost your question.
  • Using definitions and properties of inverses:
    1. As you say \"e%5Ex\" and \"ln%28x%29\" are inverses of each other.
    2. So \"e%5E%28ln%28x%29%29\" is a composition of inverses.
    3. The composition of all inverses results in the identity function: f(x) = x.
  • Using Algebra:
    1. Find the natural logarithm of each side:
      \n" ); document.write( "\"ln%28e%5E%28ln%28x%29%29%29+=+ln%28x%29\"
    2. Use the property of logarithms, \"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\" to move the exponent of the argument in front of the logarithm:
      \n" ); document.write( "\"ln%28x%29%2Aln%28e%29+=+ln%28x%29\"
    3. By definition, ln(e) = 1:
      \n" ); document.write( "\"ln%28x%29%2A1+=+ln%28x%29\"
    4. By the Identity Property of Multiplication the left side simplifies to:
      \n" ); document.write( "\"ln%28x%29+=+ln%28x%29\" which is true for all positive values of x.
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