What is the rule called where $e^{\ln {x}}=x$?

Question

I have seen this rule $$e^{\ln {x}}=x$$ used in a lot of Youtube videos, but I can't seem to find an explanation of how it works...

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Answer 1

Suppose that there is y such that e^y = x. Apply the natural logarithm on both sides of the equation. You get

ln (e^y) = ln x

y*ln (e) = ln x

y = ln x.

That's it.

Answer 2

I think learning logarithms is challenging primarily because the word "logarithm" seems strange and scary. I would prefer to call $\log_b(x)$ "the exponent that takes you from $b$ to $x$" (except that this phrase is too long to say repeatedly).

In other words, $\log_b(x)$ is defined to be the exponent such that $b$ raised to this exponent is equal to $x$: \begin{equation} b^{\log_b(x)} = x. \end{equation}

Thus, the rule in question is nothing more than the definition of $\log(x)$