How and why does $\ln (e^{0.023t}) = 0.023t$

Question

I'm so confused.

By definition, $\ln$ and $\exp$ are inverses of each other (where they make sense). So I fail to see the confusion here. — vonbrand, Mar 20 '14 at 19:04
How does one explain why one is confused? Somehow, I'd been working with ln and exp the last few months and must have passed over that rule, either going too quickly through material or just failing to make a note of it and forgot. Before I posted, I spent over an hour looking and researching. I guess I just didn't think it was easy and didn't look back far enough. What is easy/basic to one person isn't to another. Isn't what this forum is about? Tony & several others got it. — Monica, Mar 21 '14 at 04:42

score 7 · Accepted Answer · answered Mar 20 '14 at 18:43

7

It's because $\ln e^x = x$ in general. The logarithm and exponential functions are inverses of each other.

answered Mar 20 '14 at 18:43

Tony

Beat me to it by 3 seconds! +1 – Roger Mar 20 '14 at 18:44
$f(x)=e^x \Rightarrow f^{-1}(x)=\ln(x), e^{\ln x}\equiv Id$ – maxuel Mar 20 '14 at 18:50

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