# rewrite exponential function

• Sep 5th 2008, 10:53 AM
ToastE
rewrite exponential function
y=2^x

rewrite in terms of e
• Sep 5th 2008, 11:03 AM
Moo
Hello,
Quote:

Originally Posted by ToastE
y=2^x

rewrite in terms of e

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 AM
Simplicity
Quote:

Originally Posted by ToastE
y=2^x

rewrite in terms of e

Note that:

$\displaystyle a^x = \left(e^{\ln a}\right)^x = e^{x \ln a}$
• Sep 5th 2008, 11:11 AM
ToastE
Quote:

Originally Posted by Moo
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 $\displaystyle y=2^x \Longleftrightarrow y=e^{x \cdot \ln(2)}$

The only thing I am confused about is the math]e^{\ln(X)}=X[/tex]
• Sep 5th 2008, 11:13 AM
ToastE
THANK YOU
• Sep 5th 2008, 12:16 PM
Matt Westwood
Quote:

Originally Posted by ToastE
The only thing I am confused about is the math]e^{\ln(X)}=X[/tex]

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 PM
Chop 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 PM
Matt Westwood
In case you wonder, $\displaystyle \ln \left({e^X}\right) = X$ as well.