y=2^x

rewrite in terms of e

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- Sep 5th 2008, 10:53 AMToastErewrite exponential function
y=2^x

rewrite in terms of e - Sep 5th 2008, 11:03 AMMoo
Hello,

Let $\displaystyle X=2^x$

We know that $\displaystyle e^{\ln(X)}=X$

--> $\displaystyle 2^x=e^{\ln(2^x)}$

Using the rule $\displaystyle \ln(a^b)=b \ln(a)$ :

$\displaystyle 2^x=e^{x \cdot \ln(2)}$

Thus $\displaystyle y=2^x \Longleftrightarrow y=e^{x \cdot \ln(2)}$

Late edit : ok, that was my 3:D:D:Dth post ! xD - Sep 5th 2008, 11:03 AMSimplicity
- Sep 5th 2008, 11:11 AMToastE
- Sep 5th 2008, 11:13 AMToastE
THANK YOU

- Sep 5th 2008, 12:16 PMMatt Westwood

Taking logarithms is the inverse of raising to a power. So a natural logarithm is the inverse of raising to power e. It just is, that's how it's defined.

So $\displaystyle x = \ln y$ means the same as $\displaystyle y = e^x$.

what you're saying when you say $\displaystyle x = \ln y$ is: "What number do I raise e to the power of in order to get y?"

So $\displaystyle e^{\ln(X)}=X$ means "e to the power of the number you have to raise e to the power of to get X is X." - Sep 5th 2008, 12:25 PMChop Suey
To demonstrate what Matt said:

$\displaystyle \ln{a} = b $

$\displaystyle a = e^b$

$\displaystyle \therefore a = e^{\ln{a}}$ - Sep 5th 2008, 12:28 PMMatt Westwood
In case you wonder, $\displaystyle \ln \left({e^X}\right) = X$ as well.