SOLUTION: prove: e^(ln(x)) = x I know that e and ln are inverses. I'd like a rigorous proof though. thanks

Question 247675: prove: e^(ln(x)) = x
I know that e and ln are inverses. I'd like a rigorous proof though. thanks
Answer by jsmallt9(3758) (Show Source): 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 and are inverses of each other.
2. So 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:
2. Use the property of logarithms, to move the exponent of the argument in front of the logarithm:
3. By definition, ln(e) = 1:
4. By the Identity Property of Multiplication the left side simplifies to:
  which is true for all positive values of x.

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