# Is ln(√e^π)/π a rational number?

1. Sep 6, 2014

### johann1301

is ln(√eπ)/π a rational number?

where π =3.14...

2. Sep 6, 2014

### WWGD

Wow, e alone not hard-enough? You could write a whole paper, if not a small book in answering this.

3. Sep 6, 2014

### johann1301

Isn't it a half?

4. Sep 6, 2014

### WWGD

You may be right; I guess I jumped the gun.

5. Sep 6, 2014

### johann1301

but my textbook says its an irrational number, how can that be?

6. Sep 6, 2014

### PeroK

\begin{align} \frac{ \ln{ \sqrt{ e^\pi } } }{\pi} &= \frac{ \ln{ e^\frac{\pi}{2} } }{\pi}\\ &= \frac{1}{\pi} \frac{\pi}{2}\\ &= \frac{1}{2} \end{align}
The $\sqrt{e^\pi}$ is equivalent to $e^{\frac{\pi}{2}}$, so the natural log cancels with $e$ and you're left with $\frac{(\frac{\pi}{2})}{\pi}$ which is $\frac{1}{2}$.