How do you come up with $e^{\ln{x}}$

Question

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

Answer 1

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

Answer 2

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.

Answer 3

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

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

Answer 4

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: $$e^{\ln x}=\exp\circ\ln x=x$$