rewrite exponential function

**ToastE** · September 5th 2008, 10:53 AM

y=2^x

rewrite in terms of e

**Moo** · September 5th 2008, 11:03 AM

Hello,

Originally Posted by ToastE

y=2^x

rewrite in terms of e

Let $X=2^x$

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

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

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

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

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

Late edit : ok, that was my 3

th post ! xD

**Simplicity** · September 5th 2008, 11:03 AM

Originally Posted by ToastE

y=2^x

rewrite in terms of e

Note that:

$a^x = \left(e^{\ln a}\right)^x = e^{x \ln a}$

**ToastE** · September 5th 2008, 11:11 AM

Originally Posted by Moo

Hello,

Let $X=2^x$

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

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

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

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

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

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

**ToastE** · September 5th 2008, 11:13 AM

THANK YOU

**Matt Westwood** · September 5th 2008, 12:16 PM

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

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

So $e^{\ln(X)}=X$ means "e to the power of the number you have to raise e to the power of to get X is X."