$e^{\ln((9)/\log(3))}=100$? Why is that? I was messing around with my calculator and meant to put in $\log(9)/\log(3)$, which was supposed to be two. I look back, and I see my mistake, but curiously, I input that to the function $e^{(x)}$ and it comes out with $100$... Why is that? Why is $e^{\ln(9)/\log(3)}=100$?
 A: In the calculator  $\log$ is for the decimal logarithm and $\ln$ for the natural logarithm so
$$\frac{\ln 9}{\log3}=\frac{2\ln 3\times \ln 10}{\ln3}=\ln 100$$
A: Hint: 
$$\log(3)=\log_{10}(3)=\frac{\ln(3)}{\ln(10)}\,.$$
A: We can convert $\log(3)$ to base $e$ with the change of base rule:
$$\log(3) = \frac{\ln(3)}{\ln(10)}$$
Then the expression $(\ln(9)/\log(3))$ equals $\frac{ln(9)}{\ln(3)/\ln(10)}$, which is equal to 
$$\begin{align} 
\ln(9) \cdot \frac{\ln(10)}{\ln(3)} &= 2\ln(3) \cdot \frac{\ln(10)}{\ln(3)}\\
&= 2 \cdot \ln(10)\\
&= \ln(100)
\end{align}$$
So $e^{\ln(9)/\log(3)} = e^{\ln(100)}$. We know $e^{\ln(a)} = a$, so $e^{\ln(100)} = 100$.
A: To take it one step at a time, recall the definition of the logarithm.
In particular, $\ln9$ is the (unique) number that satisfies $\mathrm{e}^{\ln9}=9$.
Also, $\log3$ is defined by $10^{\log3}=3$.
In this way, $$\mathrm{e}^{\ln9/\log3}=\left(  \mathrm{e}^{\ln9}\right)  ^{1/\log
3}=9^{1/\log3}$$
and
$$9^{1/\log3}=100$$ because $$100^{\log3}=10^{2\log3}=\left(  10^{\log3}\right)  ^{2}=3^{2}=9.$$
A: Because $\ln(9)=\log(9)\ln(10)=2\log(3)\ln(10)$, so $\ln(9)/\log(3)=2\ln(10)$
A: $\log(e^{\ln((9)/\log(3))})=\dfrac{1}{\log3}\log(e^{\ln9})=\dfrac{1}{\log3}\log9=
2\dfrac{\log3}{\log3}=2$
A: So this is the equation:
$y=abs(e^{(ln(x^2)/log(x)})$
@Newb I'm sorry for taking your work, I'm not familiar with this.
We can convert log(x) to base e with the change of base rule:
$\log(x) = \frac{\ln(x)}{\ln(10)}$
Then the expression $(\ln(x^2)/\log(x)) equals \frac{ln(x^2)}{\ln(x)/\ln(10)}$, which is equal to
\begin{align} 
\ln(x^2) \cdot \frac{\ln(10)}{\ln(x)} &= x\ln(x) \cdot \frac{\ln(10)}{\ln(x)}\\
&= x \cdot \ln(10)\\
&= x\ln(10)
\end{align}
So $e^{\ln(x^2)/\log(x)} = e^{x\ln(10)}$. But when you graph this equation, it doesn't equal 100...? Am I doing something wrong? 
