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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...

(Line 4) https://gyazo.com/a84a2b6cb38f30a07e0d73aca1456642

Carser
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Anonymous
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  • Please try to avoid using links as what we are to go through. Instead, post the content from the link that you wish us to observe and then the link for if we need additional context. – Simply Beautiful Art Sep 03 '16 at 01:08
  • The rule is that $e^{\log z} = z$. In my mind, I think of this as being the definition of $\log z$. In other words, $\log z$ is defined to be the exponent that takes you from $e$ to $z$. – littleO Sep 03 '16 at 01:08
  • It's an identity from the two functions being inverse. https://en.wikipedia.org/wiki/Natural_logarithm – Carser Sep 03 '16 at 01:08
  • It's definition. By definition $\log_a x $ is the number $y $ so that $a^y= x$. So $\ln x $ is the number $y$ so that $e^y = x $. So $e ^{\ln x} = x $ by definition. – fleablood Sep 03 '16 at 03:52
  • ... although different calculus books have different methods of defining e and ln and whether to prove that ln actually is $log_e $. However it's defined, one way or another it is revealed ln = $log_b $ and that $e^{\log_e x} = x$ by definition. – fleablood Sep 03 '16 at 03:59

2 Answers2

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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.

Rodrigo Almeida
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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 $\log(x)$

littleO
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