Why does exponential(ln(x))=x?

Question

When you try to reform this equation (by taking log on both sides), you always end up with ln(x) = ln(x), so Why does exponential(ln(x))=x?

Answer 1

The exponential function $\text{exp}(z)$ is defined as:

$$\text{exp}(z):=\sum\limits_{j=0}^\infty \frac{z^j}{j!}$$

By the ratio test, this function converges for every complex value of $z$.

One can prove that $\text{exp}(z)$ satisfies $\text{exp}(a+b)=\text{exp}(a)\cdot\text{exp}(b)$

The value $e$ we define to be equal to $\text{exp}(1)$. We often shorthand the exponential function by simply writing it as $\text{exp}(z)=e^z$, and we can prove that it satisfies all of the usual properties that we expect of it.

Most importantly, is that we can show $e^x$ is strictly monotonic increasing and continuous over the real numbers, hence it forms a bijective function from its domain to its range.

Since it is a bijection, by theorem, there must exist a functional inverse. We define that functional inverse to be called $\ln$.

As the functional inverse of $e^x~:~\Bbb R\to (0,\infty)$, we have $\ln(x)~:~(0,\infty)\to \Bbb R$ and it satisfies the properties of being a functional inverse, namely $\ln(e^x)=x$ for all $x\in \Bbb R$ and $e^{\ln(x)}=x$ for all $x\in (0,\infty)$.

One may continue and further define the logarithm for negative and complex numbers. Consult any introductory textbook on Complex Analysis for further information.

Answer 2

log_e (x) =y is the number e needs to be raised to to get to x (by definition of logarithm). We call this ln.

Answer 3

By definition $\ln y $ is the power you must raise $e $ to in order to get $y$.

So what do you get when you raise $e $ to the power you must raise $e $ to in order to get $y $?

Well, you raised $e $ to that power, and that power was the power that gives you $y$. So you get $y$.

In other words.....

$e^x=y \iff \ln y =x$ because this is how $\ln y $ is defined.

So if $\ln y =x $ then $e^{\ln y} = e^x=y$.

Answer 4

The exponential function is the function which maps $x\mapsto \sum_{i=0}^{\infty} \frac{x^j}{j!}$. It is often denoted by $\exp{x}$ or $e^x$. The logarithm of $x$ is the map $x\mapsto \int_1^x \frac{1}{t}dt$ and is often denoted by $\log(x)$ (some use $\ln(x)$). It is a non obvious fact that these two functions are inverse to one another. It would not be easy to prove this by directly composing the two functions. However, if you want to prove this on your own, I do suggest trying the direct route first (if only to convince yourself that there must be an easier way).