Why logs and exponentials undo each other

Someone on the internet asks:

I don't get why $\ln(e^x) = e^{\ln(x)}$. Can you explain?

Of course I can! Or at least, I can try; the easy answer is to say 'by definition', but that doesn't help you much.

$\log_n(x)$ answers the question "what power would I raise $n$ to, to get $x$?" For example, $\log_{10}(100) = 2$, because $10^2 = 100$.

$\ln$ is the same thing as $\log_e$ , by definition. So, $\ln( e^x )$ answers the question "what power would I raise $e$ to, to get $e^x$?" -- which is clearly $x$.

Similarly, $e^{\ln(x)}$ is $e$ to the power of "whatever you raise $e$ to the power of to get $x$" -- and if you raise $e$ to that power, you get $x$.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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