0

I know that $e^{\ln{x}}$ is just the inverse of the exponential function but I don’t get it how it is done to arrive with the form: $e^{\ln{x}}=x$. Let’s say this:

$\ln{x} = a$ so $x = e^a$, but my point is how can I come up with the form $e^{\ln{x}} = x$.

Michael Rybkin
  • 6,646
  • 2
  • 11
  • 26
Bido262
  • 85
  • This is not clear. What are you asking? – lulu Apr 20 '19 at 13:11
  • $e^{\ln x}=x$ for $x > 0$ is a consequence of $e^x$ being the inverse of $\ln x$. – Randall Apr 20 '19 at 13:12
  • Sorry if it sounds unclear. My question is how do you cone up with the form e^ln(x) = x. I can pnly come up with “x = e^a”. – Bido262 Apr 20 '19 at 13:12
  • But if $a=\ln{(x)}$ as you stated then $e^a=e^{\ln{(x)}}$ – Peter Foreman Apr 20 '19 at 13:13
  • If you didn't know that exponentiation and logs were inverse functions, you could notice that $\ln e^{\ln x}=\ln x\times \ln e = \ln x$ and since $\ln$ is $1:1$ this tells you that $e^{\ln x}=x$. – lulu Apr 20 '19 at 13:13
  • Yes you’re right. But my point is how can I derive it to cone up with something like a = e^ln(a). What did I miss or I should do in order to come up with the inverse form. – Bido262 Apr 20 '19 at 13:14
  • This isn't getting any clearer. – Randall Apr 20 '19 at 13:15
  • 1
    Finally undertood the concept of lulu. Can you post it as answer so that I could mark this answered. That was what i was trying to understand. – Bido262 Apr 20 '19 at 13:16
  • But, really, one of those functions is defined (by an integral or some other process) and then the other is defined to be the inverse. Using algebraic properties of logs and exponentiation (as I did in my prior comment) is somewhat circular as you need to know that the functions are inverse to each other to justify the steps. – lulu Apr 20 '19 at 13:16
  • I don't want to post it as an answer because, as I pointed out, it is somewhat circular. The key fact is that the functions are inverse to each other. – lulu Apr 20 '19 at 13:17

4 Answers4

0

I think the answer is there in your question , which is

Let $a=\ln x\tag {1}$

so $$x=e^a \tag {2}$$ now substitute $1$ in $2$ to get $$x=e^{a=\ln x} $$

E.H.E
  • 23,280
0

You can prove that $e^{\ln{x}}$ is equal to $x$ just by using the definition of the logarithmic function. Let $e^{\ln{x}}$ be $y$:

$$ e^{\ln{x}}=y \Longleftrightarrow \ln{y}=\ln{x}\implies y = x. $$

Therefore: $$e^{\ln{x}}=x.$$

The expression $\ln{y}=\ln{x}$ says that $\ln{x}$ is the power we should raise $e$ to to get $y$ and $e^{\ln{x}}=y$ is just another way to write that.

Michael Rybkin
  • 6,646
  • 2
  • 11
  • 26
0

You have $\ln x = a$ so $x = e^a$. Then substitute:

$$ e^{\ln x} = e^a = x $$

GEdgar
  • 111,679
0

Firstly $e^{\ln x}$ is not the inverse of exponential function. The inverse of the exponential function is $x\mapsto \ln{x}$.

Secondly by the définition of inverse function, when we compose a function with his inverse we end getting the Identity function; more formally: $f\circ f^{-1}(x)=x$ Therefore: $$e^{\ln x}=\exp\circ\ln x=x$$

DINEDINE
  • 6,081