y=2^x
rewrite in terms of e
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 3th post ! xD
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."