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:

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

Answer 1

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

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

Answer 2

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