1. ## rewrite exponential function

y=2^x

rewrite in terms of e

2. Hello,
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 3th post ! xD

3. Originally Posted by ToastE
y=2^x

rewrite in terms of e
Note that:

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

4. 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]

5. THANK YOU

6. 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."

7. To demonstrate what Matt said:

$\displaystyle \ln{a} = b$

$\displaystyle a = e^b$

$\displaystyle \therefore a = e^{\ln{a}}$

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