Exponents with variables inside exponents

Question

I am confused about how to reduce this, is there any way?

$\sqrt x \ln(\sqrt x) = ln(\sqrt x^\sqrt x) = $ $?$

This can be written like this too:

$ln(x^\frac{x^\frac{1}{2}}{2}$)

Or:

$ln(x^\frac{\sqrt x}{2}$)

score 1 · Answer 1 · answered May 15 '14 at 02:49

1

$$\sqrt{x}\ln (\sqrt{x})=\dfrac{\sqrt{x}}{2}\ln x$$ If you need to simplify it more, please comment.

answered May 15 '14 at 02:49

I realize your answer is a better way to simplify it, but is there a way to simplify this: $\sqrt x^\sqrt x$ Just curious – JeremiahBarrar May 15 '14 at 02:54
You can only write it as: $x^{\frac{1}{2}\sqrt{x}}$ – Kal S. May 15 '14 at 02:56
Ok, thanks. If x represents a big equation, it would be nice to reduce it to one x variable. – JeremiahBarrar May 15 '14 at 02:58
@JeremiahBarrar $$\sqrt{x}^{\sqrt{x}}=\exp\left(\dfrac{\sqrt{x}}{2}\ln x\right)$$ – May 15 '14 at 03:23

score 0 · Answer 2 · answered May 15 '14 at 02:48

0

You have $\ln{(\sqrt{x})}=\frac{1}{2}\ln{x}$ so:

$\sqrt x \ln(\sqrt x) = 1/2 \sqrt{x} \ln{x}$

answered May 15 '14 at 02:48

Kal S.

2 Answers2